STUDENT UNDERSTANDING OF FUNCTION AND SUCCESS IN CALCULUS
by
Daniel I. Drlik
A thesis
submitted in partial fulfillment
of the requirements for the degree of
Master of Science in Mathematics Education
Boise State University
May 2015
© 2015
Daniel I. Drlik
ALL RIGHTS RESERVED
BOISE STATE UNIVERSITY GRADUATE COLLEGE
DEFENSE COMMITTEE AND FINAL READING APPROVALS
of the thesis submitted by
Daniel I. Drlik
Thesis Title: Student Understanding of Function and Success in Calculus Date of Final Oral Examination: 05 March 2015 The following individuals read and discussed the thesis submitted by student Daniel I. Drlik, and they evaluated his presentation and response to questions during the final oral examination. They found that the student passed the final oral examination. Laurie Cavey, Ph.D. Chair, Supervisory Committee Sasha Wang, Ph.D. Member, Supervisory Committee Margaret T. Kinzel, Ph.D. Member, Supervisory Committee
The final reading approval of the thesis was granted by Laurie Cavey, Ph.D., Chair of the Supervisory Committee. The thesis was approved for the Graduate College by John R. Pelton, Ph.D., Dean of the Graduate College.
ACKNOWLEDGEMENTS
I need to acknowledge my advisor, Dr. Laurie Cavey. Her continuous support,
understanding, comments and instruction through the process were priceless in my efforts
to complete this thesis. I also need to acknowledge the other members of the committee,
Dr. Margaret Kinzel and Dr. Sasha Wang, whose comments and knowledge aided in the
completion of this thesis.
I must also acknowledge my family. First to my wife for her understanding and
patience through this process. Next to my kids who, no matter how frustrated I got or
how long I was away, always welcomed me with open arms.
Finally, I would like to acknowledge the students who were willing to take part in
this study. As without them, the study would not have been able to happen.
iv
ABSTRACT
The purpose of this study was to determine if there is a relationship between
student success in calculus and student understanding of function. Student understanding
of function was measured using two questionnaires, one of which is a modification of an
existing measure based on APOS theory. The other I developed with items from the
concept image literature. The participants of this study were 116 high school students
who were enrolled in a first-year calculus course. The results of the questionnaires were
aligned to course exam scores to determine connections between function understanding
and rate of success in calculus.
A major finding of this study is that students can be successful in a first-year
calculus course without demonstrating a process level understanding of function at the
beginning of the course. In general, a positive correlation between understanding of
function and success in calculus was found.
An item-by-item analysis of the two questionnaires revealed that students
demonstrated competence, relative to their measured understanding of function, with
items that are typically presented in standard mathematics courses taken prior to calculus,
such as when provided a function as an algebraic rule and asked to calculate the value of
the function. Also, students tended to justify decisions for solutions based on criteria not
necessarily related to the definition of function. This however, appeared to have little
impact on the level of success a student was able to achieve in calculus.
v
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ..................................................................................................... iv
ABSTRACT .............................................................................................................................. v
LIST OF TABLES ................................................................................................................... ix
LIST OF FIGURES .................................................................................................................. x
CHAPTER ONE: INTRODUCTION ....................................................................................... 1
Statement of the Problem .............................................................................................. 1
Research Questions ....................................................................................................... 3
CHAPTER 2: REVIEW OF THE LITERATURE ................................................................... 5
Overview ....................................................................................................................... 5
What Does It Mean for Students to Understand Function? .......................................... 9
APOS Theory of Understanding as Working with Functions ........................ 10
Understanding in Terms of the Definition of Function .................................. 12
Other Aspects of Functions that Inhibit Understanding ............................................. 14
Developing Understanding of Function ...................................................................... 15
Functions in Calculus .................................................................................................. 17
Summary ..................................................................................................................... 19
CHAPTER THREE: METHODOLOGY ............................................................................... 21
Research Design.......................................................................................................... 21
Participants and Setting............................................................................................... 22
vi
Data Collection ........................................................................................................... 23
Questionnaire 1 ........................................................................................................... 23
Questionnaire 2 ........................................................................................................... 27
Performance in Calculus ............................................................................................. 29
Data Analysis .............................................................................................................. 30
CHAPTER FOUR: RESULTS ............................................................................................... 32
Questionnaire 1 ........................................................................................................... 32
Questionnaire 2 ........................................................................................................... 37
Composite Group Results ........................................................................................... 39
The Average Understanding Group ............................................................................ 43
The Fair Understanding Group ................................................................................... 45
The Weak Understanding Group ................................................................................ 46
Determining Success in Calculus................................................................................ 47
CHAPTER FIVE: DISCUSSION ........................................................................................... 54
Is There a Correlation Between Student Understanding of Function and Performance in Calculus? ........................................................................................... 54
What Patterns in Student Understanding of Function Are Related (or Unrelated) to Student Performance in Calculus? .......................................................................... 56
Limitations .................................................................................................................. 60
CHAPTER SIX: CONCLUSIONS ......................................................................................... 63
Implications as a Teacher ............................................................................................ 63
Future Research .......................................................................................................... 64
REFERENCES ....................................................................................................................... 66
APPENDIX A ......................................................................................................................... 70
vii
Questionnaire 1: Modified PCA (Precalculus Assessment, n.d.) ............................... 70
APPENDIX B ......................................................................................................................... 79
The Precalculus Assessment (PCA) Taxonomy Aligned With Questionnaire 1 Items (Precalculus Assessment, n.d.) .......................................................................... 79
APPENDIX C ......................................................................................................................... 83
RUBRIC for Questionnaire 1...................................................................................... 83
APPENDIX D ......................................................................................................................... 91
Questionnaire 2 ........................................................................................................... 91
APPENDIX E ......................................................................................................................... 94
RUBRIC for Questionnaire 2...................................................................................... 94
APPENDIX F.......................................................................................................................... 98
IRB Approval Form .................................................................................................... 98
APPENDIX G ....................................................................................................................... 100
Informed Consent/Assent Form ................................................................................ 100
viii
LIST OF TABLES
Table 1. Score ranges for Questionnaire 1 .............................................................. 26
Table 2. Score ranges for Questionnaire 2 .............................................................. 28
Table 3. Success in calculus based on course ......................................................... 29
Table 4. Composite score ranges ............................................................................ 30
Table 5: Percentage of students scoring an average greater than 2.5 or less than or equal to 2.5 on all items corresponding to sections of the PCA taxonomy................................................................................................... 33
Table 6: Item breakdown of Questionnaire 1 .......................................................... 34
Table 7: Item breakdown of Questionnaire 2 ......................................................... 38
Table 8: Student results on the two questionnaires ................................................. 39
Table 9: Composite score category percentages (based on scores on two questionnaires) .......................................................................................... 40
Table 10: Subgroup performance on Questionnaire 1 taxonomy categories ............ 43
Table 11: Student breakdown by calculus course. .................................................... 47
Table 12: Basic Statistics on Exam Scores ............................................................... 48
Table 13: Successful and not successful student percentages at each level of understanding of function. ........................................................................ 50
Table 14: Correlation between average exam scores and questionnaire scores by course. ....................................................................................................... 52
Table 15: Linear Regression models for statistically significant regression data. .... 53
ix
LIST OF FIGURES
Figure 1. APOS Constructions for mathematical understanding (Asiala et al., 1996) ........................................................................................................... 8
Figure 2. PCA question modeling the velocity of two cars from t = 0 to t = 1 hr. ... 14
Figure 3: Graph of Student Results of Questionnaire 1 vs. Questionnaire 2 ........... 41
Figure 4: Distribution of exam scores for students enrolled in AP Calculus AB..... 48
Figure 5: Distribution of exam scores for students enrolled in Introduction to Calculus..................................................................................................... 49
x
1
CHAPTER ONE: INTRODUCTION
Statement of the Problem
As a calculus teacher, I expect students to come to me with conceptual
understanding of functions. For example, students should be able to solve equations,
graph and evaluate functions, transfer between forms of functions and compute basic
sums, differences, products and quotients of functions. However, this is far from the
truth. As I introduce topics to students in calculus, I find myself reviewing topics from
previous courses. For this reason, there is often much frustration for the teacher as well as
the student. My frustration stems from having to find time to review topics that are not
necessarily part of the curriculum. The students become frustrated due to their struggles
in trying to learn new topics without having complete understanding of foundational
material. Identifying the area in which a student struggles and correcting the way a
student understands a topic should be a goal at each level of mathematics.
In a first-year calculus course, functions play a critical role. With an incomplete
understanding, a student will tend to struggle. Students need to be able to shift from one
representation of function to another as called for by the problem at hand. Every year,
this deficiency is almost immediately recognized in calculus as we begin the year. The
curriculum that I teach starts with limits where students are trying to identify a y-value
that a function approaches as the x-value gets arbitrarily close to a particular quantity.
With this topic students are immediately tested on their ability to shift between the
different representations of function, namely equation, graphical, tabular, and description
2
of a contextual situation. There are times when students are asked to use a graphical
representation of function to evaluate a limit. If the representation of function does not fit
into what students understand about what a function should look like, for example, the
graph of a piece-wise defined function, some students produce an incorrect limit. Other
times students have to calculate limits given a function defined by a rule or equation.
Here students are comfortable evaluating limits by substitution, but struggle when the
limit could not be evaluated in this way. Many students appear to understand these
concepts as I introduce them through examples, however, once they are asked to perform
them on their own, they are unsure which technique or procedure they should implement.
This issue is not only faced with limits, but throughout the calculus curriculum. A
possible cause for student difficulty with functions is described in the following quote:
“If algebraic and procedural methods were more connected to conceptual learning,
students would be better equipped to apply their algebraic techniques appropriately in
solving novel problems and tasks” (Oehrtman, Carlson, & Thompson, 2008, p. 28). That
is, the difficulties described above stem from the fact that students are being introduced
to procedures rather than the concept.
Aside from the use of the appropriate procedure or representation, what students
hold as a definition of function can play a critical role in their success when interpreting
or working with functions. Students frequently identify a polynomial in the form,
, where a, b, and c are rational numbers, as a function but when
it comes to other functions like trigonometric, radical, piece-wise or rational functions,
their definitions possibly lead to incorrect interpretations of the problem. An example of
this occurs when thinking about the domain, or values for which an expression is defined,
cbxaxxf nn +++= − ...)( 1
3
such as . Students frequently have difficulty identifying the domain as all
real numbers 𝑥, such that . This is an important skill to have in calculus, as students
should only interpret a function using differentiation, integration and other calculus
techniques for values in the domain of that function.
Research Questions
My observations as a calculus instructor at the high school level have prompted
me to more carefully investigate how student understanding of function may be related to
student success (or lack thereof) in a first-year calculus course. Based on the research
literature student difficulties with understanding functions can be due to several factors
throughout the development in the courses that the students have taken prior to calculus.
Students need to know how to work with a function in order to solve the problem at hand
(see for example Breidenbach, Dubinsky, Hawks, & Nichols, 1992, and Oehrtman et al.,
2008). They also need to be able to make connections between different representations
of functions, that is: tabular, graphical, equation form and word form (see for example
Carlson, n.d., Dubinsky & Wilson, 2013 & Davis, 2007). Researchers have also
examined issues related to the development of functional understanding and the
importance that functions play in calculus (see for example Monk, 1994; Carlson, 1997;
& Oehrtman et al., 2008). For example, Oehrtman et al. (2008) suggest that students need
a strong understanding of functions in order to be successful in calculus.
In this study I aimed to determine if students with a deep understanding of
function as they enter calculus tended to perform better in calculus than those students
1)( −= xxf
1≥x
4
with a lesser understanding of function at the beginning of the course. The following
research questions were used to guide my work.
1. Is there a correlation between student understanding of function and performance
in calculus?
2. What patterns in student understanding of function are related (or unrelated) to
student performance in calculus?
5
CHAPTER 2: REVIEW OF THE LITERATURE
As students enter high school mathematics, functions play a vital role in students’
studies. By the time a student reaches more advanced mathematics classes, like calculus,
they must be comfortable working with functions and be able to apply the definition of
function. In this chapter, I define what it means for a student to understand function,
student ideas that inhibit understanding of function, and what teachers can do to develop
student understanding of function. First, I provide an overview of the ideas presented in
this chapter.
Overview
In a first-year calculus course students are introduced to the concepts of limits,
differentiation, and integration. However, as Monk (1994) states, “the concept that the
subject is built out of, the one that lies behind such notions as limit, derivative, and
integral is that of function” (p. 21). In students’ learning of calculus they work with a
variety of functions, which include: polynomial functions, rational functions,
trigonometric functions, piece-wise functions, transcendental functions, and radical
functions. This leads to an interesting question. What role does understanding of
functions play in a student’s success in calculus?
According to the Idaho Common Core State Standards (ICCSS; Common Core
State Standards Initiative, n.d.) the term function is introduced as early as the eighth
grade. Students have also been exposed to functions in courses prior to calculus, like
6
precalculus. Therefore, one might expect a student who enters higher level math courses
in high school, like calculus, to know and apply the definition of function, know the
purpose that a function serves, and know how to manipulate and interpret functions.
Formally, a function in mathematics is defined as “a correspondence between two
nonempty sets that assigns to every element of the first set (the domain) exactly one
element in the second set (the codomain)” (Vinner & Dreyfus, 1989, p. 357). This
definition of function is known as the Dirichlet-Bourbaki concept (Vinner & Dreyfus,
1989) and allows for functions that are discontinuous, functions defined on split domains
and functions defined by graphs or ordered pairs, and so on.
In general, students enter the classroom every year with different levels of
understanding of function. Mason (2008) states that teaching cannot force, necessitate or
guarantee learning, but teaching can make learning more likely and more effective if it
makes use of learners’ powers and dispositions, and exposes them to significant and
fruitful ways of thinking and perceiving (p. 271). In the case of functions, a student
introduced to functions as simply a tool to manipulate numbers may be limited to this
idea throughout his or her mathematical studies. The student who is introduced to
function in its many representations and purposes may have a better understanding.
Because of its relevance to so many other mathematical topics and its role in
college-level mathematics, function constitutes one of the most important topics in
secondary school mathematics (Cooney, Beckman, & Lloyd, 2010, p. 7). As noted in the
National Research Council’s 1989 report, Everybody Counts, “if undergraduate
mathematics does nothing else, it should help students develop function sense” (p. 51).
From the early introduction in middle school to college-level mathematics courses,
7
students are using, analyzing and interpreting functions. Cooney et al. (2010) list a few
examples of textbook definitions of function. These include:
• A function is a relationship between input and output. In a function, the output
depends on the input. There is exactly one output for each input.
• A function is a relation in which each element of the domain is paired with
exactly one element of the range.
• A function is a set of ordered pairs (or number pairs) that satisfies this condition:
There are no two ordered pairs with the same input and different outputs.
• A real-valued function f defined on a set D of real numbers is a rule that assigns to
each number x in D exactly one real number, denoted by f (x).
The variety of definitions found in textbooks is due to the mathematics level of
the student. Students need to be introduced to the concept of function in line with the
formal definition without being overburdened with notation and vocabulary. While
students are introduced to a variety of formal definitions of function, each student
possesses his or her own concept image of function (Vinner, 1992) and students must be
able to decipher and make sense of these formal definitions. Concept image as defined by
Vinner (1992) is a nonverbal entity associated in our mind with the concept name. A
student’s concept image of function is shaped possibly by the various definitions that
they encounter, examples that teachers use in class, or class assignments. With their
concept image, students create their own definition of function, which at times is
incorrect and therefore leads students to make incorrect interpretations (Vinner &
Dreyfus, 1989).
8
As students make progress through the mathematics curriculum, they create and
refine their concept image based on problems they are faced with in the particular class.
Breidenbach et al. (1992) and Carlson (n.d.) suggest that early in the mathematics
curriculum the aim should be to move students from viewing functions as a physical
manipulation of objects, to interiorization of objects and finally encapsulation of the
process in its totality. Figure 1 illustrates the process through which a student develops
concepts, like function, in mathematics. In the case of function, students first act on them
at the action level where a function is a tool to manipulate a number.
Figure 1. APOS Constructions for mathematical understanding (Asiala et al.,
1996)
The student is then able to interiorize this and move to the process level. At this
level a student understands that a function will produce a value, but is not concerned with
producing a specific output. From this point the student can view functions at the object
level where they have the ability to perform operations on functions. Finally, they are
able to view functions at the appropriate level as called for by the problem. Here they are
determined to be at the schema level. The terms action, process, object and schema form
the APOS framework and will be discussed further later in this chapter (Asiala et al.,
1996).
9
Student difficulty with functions occurs for various reasons. Ronda (2009) and
Clement (2001) state that representations of functions are a stumbling point for students.
These representations include equation, graphical, tabular, and descriptions of contextual
situations. Being introduced to concepts using different representations has been shown
to help students make connections between the different representations of functions
(Habre & Abboud, 2006). When introduced to concepts with different representations
students have made connections between the representations. Introducing concepts with
different representations of functions also improve students’ concept image (Vinner,
1992) and definition of function.
Because functions play a vital role in today’s mathematics curriculum and are also
so extremely useful in capturing aspects of real-life phenomena, students must develop a
deep understanding of function to ensure success in mathematics (Oehrtman et al., 2008;
Davis, 2007; Breidenbach, et al., 1992; Carlson, 1997).
What Does It Mean for Students to Understand Function?
Students need to be able to apply tools and techniques to demonstrate their
understanding of mathematical topics. For my study, I chose to focus on two aspects of
function understanding: 1) how students work with functions given in various forms, and
2) how students apply the definition of function. Whether it be reciting a definition, being
able to evaluate a function for particular values, generating various representations of a
real-life situation, or different ways of working with functions, each of these contribute to
determining a student’s level of understanding. For me, student understanding of
functions depends on the APOS level at which a student can work with functions and
how well the student can apply the definition of function.
10
Sierpinska (1992) states that understanding a concept, like functions, is achieved
when one is able to use, identify, apply, generalize and create extensions with its use. In
a study conducted by Ronda (2009), 444 students from Philippine schools in grades eight
through ten participated. Each student completed tasks that identified how well a student
understood functions in equation form. Each task allowed for the identification of growth
points, which included: equations are procedures for generating values, equations are
representations of relationships, equations describe properties of relationships, and
functions are objects that can be manipulated and transformed. Ronda concluded that the
framework, or growth points, would be a way to guide teachers and their instruction to
aid in student understanding of functions.
APOS Theory of Understanding as Working with Functions
APOS refers to the phases that a student goes through in the process of
understanding functions, action-process-object-schema (APOS; Mahir, 2010). Several
researchers describe student understanding of functions using APOS theory (Martinez-
Planell, Gaismann, 2012; Mahir, 2010; Thompson, 1994). Piaget’s mental constructions
for learning mathematics served as a guide in the development of APOS (Asiala et al.,
1996). APOS theory emerged in an attempt to interpret the type of thinking that occurs as
individuals develop understanding of mathematics, in general. A small group of
researchers have “been using APOS theory within a broader research and curriculum
development framework” (Dubinsky & McDonald, 2001, p. 4). See the list below for a
description of each phase. The phases below are ordered from the lowest level of ability
of working with functions to the highest.
11
• Action (A) View: In the action view functions are regarded as static. A
function is tied to a specific rule, formula or computation and requires the
completion of specific computations and/or steps (Oehrtman et al., 2008). A
student at the Action level is unlikely able to solve a situational problem that
involves a function without being given a formula (Moore, 2012).
• Process (P) View: At the Process View functions are regarded as dynamic. A
function is a generalized input-output process that defines a mapping of a set of
input values to a set of output values (Oehrtman et al., 2008). When working
with functions, students are at the process level if the function is not restricted
to numbers for the domain and ranges of the function, or if the student can
imagine certain operations with functions with no explicit formula
(Breidenbach et al., 1992). To better develop the understanding of the function
as a process teachers can apply reverse-path-development (Eisenberg, 1992).
Here the process is discussed in both directions. For example, a student given
two functions, f(x) and g(x), can produce the composition, f(g(x)). At the same
time if given f(g(x)) and f(x) the student is able to identify g(x).
• Object (O) View: At the Object View function is regarded as an object that can
be manipulated and changed much like any number. An object is constructed
from a process when the individual becomes aware of the process as a totality
and realizes that transformations can act on it (Dubinsky & MacDonald, 2001).
• Schema (S) View: At the schema view the student is able to transition between
the three previous views (action, process or object) as deemed necessary by the
problem. For example, when working through problems in a calculus course,
12
they may need to use the action view of a function to determine values and in
the same problem treat the function as an object in order to manipulate the
function into a usable form.
Determining the level at which a student is able to work with function is one way
to determine the level of understanding a student has of function (Dubinsky &
MacDonald, 2001).
APOS theory has been used to develop several different studies. For example, the
development of the Precalculus Concept Assessment (PCA), a validated instrument
which measures a student’s understanding of functions based on the APOS perspective
(Carlson, n.d.) measures a student’s ability to work with functions at the process level of
the APOS framework. Dubinsky and Wilson’s (2013) analysis on the effectiveness of the
Algebra Project and student understanding of function also applied APOS Theory.
Finally, Oehrtman et al. (2008) state the importance of the process view to understand
calculus and also describe how to foster the process view. The APOS framework will be
used in this study to analyze the aspect of student understanding of function related to
how students work with functions.
Understanding in Terms of the Definition of Function
Definitions also play a vital role in student understanding of functions. Students
build the strongest definitions through experiences and involvement. Definitions should
only be introduced once examples of student experiences of function have been applied
and extended (Vinner & Dreyfus, 1989). That is, students should not be introduced to
definitions until they have developed an appropriate concept image. Class discussion on
the definition of functions may have a greater impact on student understanding and a
13
willingness to accept definitions if the students are involved in the creation of definitions
(Edwards & Ward, 2008). Also, Vinner (1992) claims that appropriate pedagogies should
be implemented before suggesting definitions to the students. In other words, educators
need to allow students the opportunity to, for lack of a better word, “play” with a concept
before they burden them with definitions. By the time students reach a first-year calculus
course, they should be able to apply the Dirichlet-Bourbaki definition of a function.
Thompson (1994), Sierpinska (1992), Carlson (1997) and Clement (2001) all state
that students tend to believe that a function is a mathematical statement with an equal
sign. From prior experiences with functions, students believe that the function must be
continuous and cannot be constant or defined over split domains (Dubinsky & Wilson,
2013; Sierpinska, 1992; Clement, 2001). Dubinsky and Wilson (2013) continue
discussing student difficulty with functions. Included in the discussion is the one-to-one
property. Here students think that each element of the domain must be mapped to a
different element of the range. This causes difficulty in understanding the idea of the
constant function, a function where every element in the domain is mapped to the same
element in the range. Also mentioned is the vertical line test, a test used on graphs of
functions where a vertical line is drawn to help determine whether a graph represents a
function. Because of these difficulties students’ concept images of function often have
flaws. In order for students to have a deep understanding, they must be able to interpret
features from different representations of functions as well as learn and understand the
formal definition of function (Carlson, 1997).
14
Other Aspects of Functions that Inhibit Understanding
Difficulty with interpreting graphs of functions also plays a role in student
understanding of functions. For example, students sometimes think an oscillating velocity
versus time graph means that the object is travelling over some sort of oscillating, “hilly”,
terrain rather than simply speeding up or slowing down (Monk, 1992; Oehrtman et al.,
2008). Another example is seen on the Precalculus Concept Assessment (PCA). Figure 2
illustrates the velocities of two cars travelling in the same direction. Carlson (n.d.)
remarks that there are students who interpret the graph as a collision between the two cars
occurring at 1 hour.
Figure 2. PCA question modeling the velocity of two cars from t = 0 to t = 1 hr.
Creating connections between various representations of functions is another
important aspect of understanding functions (Thompson, 1994; Davis, 2007; Carlson,
Oehrtman, & Engelke, 2010; Akkus, Hand, & Seymour, 2008) and related to another
difficulty students experience (Dubinsky & Wilson, 2013). Students have great difficulty
making the connections between various representations of functions, these include:
equations, graphs, tables and word forms (Thompson, 1994; Davis, 2007; Carlson et al.,
2010; Akkus, Hand, & Seymour, 2008). That is, if presented with the graph of a function
and asked to find the input value that produces a particular output value, a student may
15
demonstrate difficulty. Here the student is demonstrating difficulty connecting the
information given by the graph to solve a problem given in equation form.
However, representations of functions play a vital role in understanding functions.
Eisenberg (1992) and Mahir (2010) stress the visualization of a graphical representation
of function. What does the picture look like? If students are able to visualize a function,
concepts like slope, rates of change, and area would make much more sense to students
(Oehrtman et al., 2008). In other words, being able to use a mental image of the function
to understand where a function is increasing or decreasing, or where the function is
constant will aid in the student’s interpretation.
Developing Understanding of Function
Students depend upon prior knowledge, or their concept image, to interpret
meanings of mathematical concepts (Vinner & Dreyfus, 1989). If this concept image is
faulty, then it is reasonable to expect limitations associated with a student’s ability to
apply the definition. Teachers must be able to identify holes in student understanding and
aid in filing those holes (Ronda, 2009). Unfortunately, teachers frequently introduce
topics with an assumption that students enter a class with required knowledge of a topic.
This is often not true (Lobato, 2008). Introducing topics in mathematics like functions at
an appropriate time in a student’s understanding of mathematical concepts and definitions
will create more success in understanding for students (Sierpinska, 1992).
Connections that students make with mathematical concepts to previously
developed ideas also aids in the understanding of the concepts. To aid in student
understanding, Oehrtman et al. (2008), motivated by the APOS theory, state that
educators need to allow students to explain the behavior of functions using appropriate
16
terminology such as: input, output, dependent and independent variables. An example of
this in the early stages of developing the function concept could have students develop
functions that calculate the amount of money in their bank account with a job that pays a
particular amount of money. To begin moving away from just an action view, the
discussion could then progress to determining the independent and dependent variables.
This could then be extended and students could be asked to determine how long it would
take them to pay for an item they would like to buy and also interpret the graph of their
function in the context of the problem. The researchers above also mention that teachers
need to be flexible in allowing students to use multiple representations. Representations
of functions should allow students to see that a function is not only a rule but rather can
also be represented as a statement or a graph. In the earlier example, students could
model the amount of money in their bank account graphically, verbally, algebraically or
in a table and observe the effects of changing aspects of the function. In this way,
students might begin to develop a process view of functions. A final remark made by the
researchers is that discussing with students not only how a function manipulates a
number, but also what is causing this change will support the development of
understanding of function. Davis (2007) and Mahir (2010) also make the argument that
the connection to real-life scenarios plays an important part for student success in the
understanding of functions. Creating a reason and connection for students not only makes
a lesson meaningful to them but it develops understanding.
Student difficulty with functions may stem from various points in their
educational progress. Kinzel (2006) conducted a study to determine if students’ abilities
to shift between the object view and process view was a reason for their difficulty to
17
work with algebraic expressions. In particular, the study involved six participants
enrolled in a mathematics education class and Kinzel interviewed the students using
questions that either involved using the process view or the object view. The process
view of an algebraic expression means an algebraic expression is being used to
manipulate a number, whereas the object view is treating the expression as an object or
quantity. The researcher’s conclusion claimed that successful students are able to
seamlessly shift between the two views.
Following a sequence of learning steps, like those of APOS, creating connections
to real life and making lessons meaningful for students are all techniques that aid in the
development of understanding. Sajka (2003) states what we usually write and do in a
mathematics lesson is very important for the student (p. 247). For example, when
students are consistently introduced to functions as a rule or equation, they tend to have a
difficult time understanding that a table or a graph of distinct points can also represent a
function (Sajka, 2003). Hence, lessons that allow for connection to real-life and also to
different representations, allow for flexibility, and take into account students’ prior
knowledge are lessons that would be the most beneficial in enhancing understanding.
Functions in Calculus
Prior to calculus, students are introduced to procedures for evaluating functions
and solving equations. Therefore, they are seeing functions as a static entity. In other
words, although they are able to view functions in the way best suited for the problem,
they are primarily instructed on functions at the action view of the APOS framework
(Carlson et al., 2010). If students have developed a deep understanding of functions prior
to attending a high school calculus course or even college, they will be better prepared for
18
these courses (Akkus et al., 2008). In calculus, functions play a vital role. Some of the
problems that students are faced with include slope of a function and the equation of
tangent lines. Because of this, development of the function concept in earlier mathematics
courses is vital for student success (Davis, 2007). Further, the ability to apply the
definition of function is a vital part of the background of any student hoping to
understand calculus (Breindenbach et al., 1992; Carlson, 1997). Much of the success that
students achieve in calculus classes appears to stem from their understanding of
functions.
Almost every aspect of a first-year calculus course depends on students’
understanding of functions. Finding limits of functions forces students to interpret
behavior of a function near a particular x-coordinate. Students must possess a dynamic
view of function, and they must view functions at least at the process level (Carlson,
1997). Carlson (n.d.) suggests that a deep understanding of the concept of function is a
vital part of the background of any student hoping to comprehend calculus. Being able to
visualize, make the connection between the equation form and the graphical form of a
function, plays a key role in understanding limits of functions. Habre and Abboud (2006)
also state that one fundamental change that calculus witnessed in recent years is an
increased emphasis on visualization. Once students begin exploring the concept of
derivative, they then explore deeper into their understanding of function. Here they are
calculating rates of change, slopes of tangent lines, looking at related rates and use the
derivative concept to describe the graphical representation of a function. Understanding
functions and being able to connect their different representations as well as being able to
19
analyze and make use of particular parts of functions is necessary for success with
derivatives.
The concept of function is central to a student’s ability to describe relationships of
change between variables, explain parameter changes, and interpret and analyze graphs
(Clement, 2001, p. 745). The final topic of a typical first-year calculus course is the
integral. Here students use integration to calculate areas of regions between two
functions, find volumes of solids created by revolving a region around an axis, and apply
integration to solve physics problems. Creating the connection of what role functions
play in each of these topics is a struggle for many students. With a strong mathematical
background and understanding of function, students are typically successful
understanding these concepts. Students with weak understanding of function struggle to
make sense of much of what happens in calculus (Oehrtman et al., 2008).
The role of functions in first-year calculus is tremendous. Ensuring that students
entering calculus or any college level mathematics course have a strong understanding of
functions is vital to their success. Every aspect of first-year calculus involves students’
understanding of function. Whether it be looking at limits, finding derivatives or using
the integral to calculate areas of regions, functions play a role somewhere in the process.
Summary
As students move on to a higher level of mathematics classes, the role of
functions and understanding of functions becomes more important. Understanding of
functions in my study is determined, in part, by a student’s ability to work with functions
as measured by the APOS framework. The ability to apply the definition of function was
used to determine another aspect of that student’s understanding of function. Calculus
20
students must enter the classroom being able to connect between the different
representations of functions, interpret and make sense of the definition of function,
manipulate functions, and understand how functions apply to phenomena in the world
around. Previous research suggests that functions must be taught using a variety of
representations and to allow students to create meanings and connections to the world
around them. Students must develop a strong definition of function and be able to
navigate through the different phases of APOS when working with functions. Doing this
will ensure readiness when beginning to analyze functions in Calculus.
21
CHAPTER THREE: METHODOLOGY
The purpose of this study was to determine whether or not there is a relationship
between student understanding of functions and their performance in calculus. Data were
collected via two questionnaires given at the beginning of the course, along with the first
semester calculus cumulative exam grades. In this chapter, I describe the research design,
the participants, and the methods for data collection and analysis.
Research Design
This is a descriptive research study. Data were gathered from two questionnaires
along with students’ semester grades from a first-year calculus course. The questions that
guided this research are:
1. Is there a correlation between student understanding of function and performance
in calculus?
2. What patterns in student understanding of function are related (or unrelated) to
student performance in calculus?
Two questionnaires (see Appendix A and D) were designed for this study to
determine student understanding of function. Questionnaire 1 is a modification of the
Precalculus Concept Assessment (Precalculus Assessment, n.d.) while I created
Questionnaire 2 inspired by work by Vinner and Dreyfus (1989). I used Questionnaire 1,
Measurement of Ability to Work with Functions to determine students’ level of
understanding based on the APOS framework. I used Questionnaire 2, Measuring Student
22
Concept Image (Definition) of Function to measure a student’s ability to apply the
definition of function. Individual student responses to the questionnaires were combined
to form a composite score to represent a student’s overall level of understanding of
function. A student’s performance in calculus was measured using the cumulative
percentage that they achieved on the exams taken in the first semester of the first-year
calculus course.
Participants and Setting
Participants of this study were members of two different high school calculus
courses taught at a large suburban high school. Of the 116 students who participated, 60
were enrolled in AP Calculus AB and 56 were enrolled in Introduction to Calculus. The
participants were chosen for convenience and ease of data collection as the researcher
had access to all who might participate in the research. All the students, in either group,
are considered higher achieving math students and thus are expected to have a deep
understanding of function.
Both calculus courses, AP Calculus AB as well as Introduction to Calculus, offer
college credit. Material that is covered in each course is relatively similar. However, AP
Calculus AB goes deeper into calculus concepts. For example, AP calculus students are
responsible for knowing how to integrate and differentiate polynomial, rational, radical,
trigonometric and other transcendental functions. However, in the Introduction to
Calculus course, students learn to differentiate and integrate polynomial, radical and
rational functions along with a few transcendental functions like exponential and
logarithmic functions.
23
From AP Calculus AB, students progress on to Calculus 2 or AP Statistics.
Introduction to Calculus students have AP Calculus AB or AP Statistics as options the
following year. Students in Introduction to Calculus are primarily seniors. Very few of
the students taking Introduction to Calculus take AP Calculus the following year.
Students enrolled in AP Calculus or Introduction to Calculus are typically eleventh or
twelfth graders. Occasionally a tenth grade student may be enrolled in AP Calculus AB.
Data Collection
Two questionnaires were administered at the beginning of the school year in
which the students were enrolled in the either of the first-year calculus courses. The
research design and questionnaires were approved through IRB (Appendix F) and
consent was collected through a form (Appendix G) sent to all students’ parents
electronically. Data from students who did not submit a consent form were not included
in the study. The questionnaires were administered by either a teacher or an
administrator at the high school. Questionnaire 1 was administered during the first two
days of the school year; Questionnaire 2 was administered one week following
Questionnaire 1. Exam grades were collected at the end of the semester for all sections of
both courses. Codes were assigned to each student by the co-principal investigator (my
advisor) to maintain confidentiality.
Questionnaire 1
Questionnaire 1 had 22 items (2 items had two parts) and was based on the
Precalculus Concept Assessment (Precalculus Assessment, n.d.), which was modified for
this research project. The PCA was chosen as a model because it is a validated instrument
24
and a good measurement of preparation for calculus (Carlson et al., 2010). The
assessment was modified from a multiple choice format to primarily an open-ended
format to discourage guessing as well as to see student thought since interviews were not
conducted. Also one of the 23 items was omitted as the wording of the question was
determined to be poor and there were other items that measured the same skill. The PCA
taxonomy (see https://mathed.asu.edu/instruments/PCA/pcataxonomy.shtml) is a listing
of reasoning, conceptual and analytic abilities that the assessment measures. I aligned
each item with the PCA taxonomy (See Appendix B) and used the categories of the
taxonomy to identify patterns in student understanding of working with functions.
Once students completed the questionnaire, each item was assigned a point value
based on a rubric that I developed (Appendix C). Point values for each item were
determined based on a combination of answers provided for the PCA and answers given
on the pilot version of Questionnaire 1 given at the end of the prior school year. I used
the discussion for each answer provided on the PCA website as well as answers provided
through the pilot version of the questionnaire to determine the point values for each item.
The example below shows how the rubric was applied to possible student responses from
item 11b.
Item 11b:
x f(x) g(x)
-2 0 5
-1 6 3
0 4 2
1 -1 1
2 3 -1
3 -2 0
25
Given the table above determine .
Score of 4: A process level of ability to work with functions would have to be
shown. Here a student would have to correctly interpret what is being asked and produce
the correct value, which is 2.
Score of 3: A student would have to show that they are at least approaching the
process level. The student shows the ability to correctly interpret the procedure involved
in the problem, however makes a mistake, for example finds )1(1 −−f rather than
. Here a student would produce an answer of 1.
Score of 2: A student demonstrates that he or she is at the action level. The
student shows the ability to use some of the given information to find a solution. Here a
student sees that the input is -1 and finds the output of g for this input, disregarding the
inverse notation. The answer that would be provided here is 3.
Score of 1: Here a student does not demonstrate the ability to interpret the
problem nor use any of the given information to find a solution. A student may simply
pick a number from the table, not listed in the above scores, as a solution.
Once the questionnaire was completed, the co-principal investigator and I scored
ten completed questionnaires. Both researchers then compared the scoring on each item
and discrepancies in scoring were discussed and corrected. Once the rubric and scoring
were agreed upon, the two researchers divided the completed questionnaires into two
parts, and each of the researchers scored a particular set of items (myself: items 1-13, co-
)1(1 −−g
)1(1 −−g
26
principal investigator: items 14-22). The co-principal investigator then recorded all data
into an Excel spreadsheet.
Points were added and the total score determined whether a student is working
with functions at the pre-action, action, pre-process or process level of the APOS
framework (Table 1). The score ranges were determined by first taking the highest
possible score of 96 (24 × 4) and by allowing for a few scores of three, the cut-off of 90
was determined. The low-end was then determined in much the same way. I included the
condition of no fours since scoring a four even on one question would show some ability
of working with functions. The two middle ranges were created to ensure that scores of
all 3’s or all 2’s would fall into the appropriate ranges of pre-process or action
respectively.
Table 1. Score ranges for Questionnaire 1
Total Score Range
90-96 60-89 38-59 24-37 (with no 4’s)
Student is at the Process Level of working with functions
Student is at the Pre-Process Level of working with functions
Student is at the Action Level of working with functions
Student is at the Pre-Action Level of working with functions
The questionnaire only allowed the researcher to determine that a student is at
most at the process level. This was determined to be acceptable as research indicates that,
first, being at the process level is important for a student to be successful in calculus
(Carlson, 1997). Second, it is very difficult to measure and distinguish the difference
between the object and process levels (Asiala et al., 1996).
27
Questionnaire 2
Questionnaire 2 consisted of eight items that I created based on work by Vinner
and Dreyfus (1989) and its intent was to determine how well a student can apply the
definition of function. The first five items of the questionnaire gave students a model (i.e.
representation) and asked students to determine whether the model is or is not a function
and then give a reason for their choice. The items included in the questionnaire were
based on items from the literature (Vinner & Dreyfus, 1989, Mahir, 2010). The models
that students were given in these first five items include: equation form, table form, graph
form, and contextual form. The next two items asked students to define a function based
on given criteria (Vinner & Dreyfus, 1989). One item asked students to define a constant
function, the other asked for a piece-wise defined function. The final item asked students
to give their definition of function. I developed a rubric (Appendix E) to assign points (3,
2, or 1) to student responses for each question. In order to score a three, the student must
have demonstrated a complete ability to use the definition of function to justify his or her
reasoning. A score of two was given when the student showed partial ability to use the
definition of function. A score of one was assigned if the student incorrectly answered the
item or if the justification showed no understanding of the definition of function.
An example of the rubric in use for item 3 is listed below. The sample responses
below show how item 3 from the questionnaire was scored.
Item 3: Is “The amount of money earned at a job.” a function? Give a reason for your answer. Score of 3 response: “It models a function.” Reason: “At different points in time you earn a certain amount of money.” Score of 2 Response: “It models a function”. Reason: “Every x has 1 y.”
28
Score of 1 Response: “It does not model a function.” Or models a function with a reason like “steady slope”.
To discuss the rationale for the scoring above, I make the following arguments.
The student whose answer earned the full 3 points showed the understanding that a
function must have one corresponding output for each input. Here the student showed this
understanding and also related the input and output to the context of the problem. The
student who earned 2 points on this item showed that they have a limited ability to apply
the definition of function. This student’s response indicates that the student thinks that x
is always an input and y is the associated output and does not show the ability to apply
the definition to a particular context. A score of 1 was earned for an incorrect answer or a
justification that shows an incomplete ability to apply the definition of function.
Scores from each of the items on Questionnaire 2 were totaled and a strength of
definition score was assigned to each student (see Table 2). The ranges below were
determined in much the same way as those for Questionnaire 1. I started with a perfect
score (all 3s) and allowing for 2s, the Strong definition range was created. A score with
all ones created the low end (Weak), then allowing for a majority of 1s with some 2s
created the range. The remaining values were then assigned to the Average range
ensuring that scoring all twos fell in that range. A student falling into each of the
categories would earn scores primarily showing that ability level (strong = 3, average = 2,
weak = 1).
Table 2. Score ranges for Questionnaire 2
Strength Strong Average Weak
Score range 21-24 13-20 8-12
29
Performance in Calculus
Students were organized in a successful category or unsuccessful category based
on the percentage of the total cumulative score they earned on all the exams taken in the
first semester. Table 3 shows the percentage necessary to be considered successful in
each of the respective classes involved in this study. The exam scores were chosen
because I felt that I had control over the scores rather than students being able to get help
on take-home assignments like homework. Also, at times the overall grade, which
includes homework, quizzes and exams may be inflated due to factors beyond my
control. The responses of students with similar grades were then compared and it was
noted as to whether any patterns emerged. For example, did low achieving students have
a weak definition of function?
Table 3. Success in calculus based on course
Course Successful Student Exam Percentage Score
Unsuccessful Student Exam Percentage Score
AP Calculus AB > 88% of possible exam points
≤ 88% of possible exam points
Introduction to Calculus > 91% of possible exam points
≤ 91% of possible exam points
I decided to use different exam percentages based on the class that the student was
enrolled in primarily because of the material that is covered in each course. In AP
Calculus AB, the students immediately begin covering calculus topics from day one.
There is no review of any algebraic concepts. Whereas, the curriculum for Introduction to
Calculus begins with an intensive review of algebraic concepts such as: domain,
functions, solving equations, and simplifying expressions. This review is covered in a
chapter and a half, therefore one and a half of the exams cover review topics on which the
30
majority of students should be successful. Also, the material in the Introduction to
Calculus course is not as in depth and thorough as it is in the AP Calculus course.
Data Analysis
To determine student understanding of function an item-by-item analysis was
done on both questionnaires as well as a comparison of the overall scores on each of the
questionnaires. I then used the responses to the questionnaires to calculate a composite
score to represent a student’s overall understanding of function. To find the composite
score I found the sum of the scores of the two questionnaires and then aligned the scores
by student. The data for students who did not answer both questionnaires were not
included.
Table 4. Composite score ranges
111-120 73-110 50-72 32-49
Strong understanding of function
Average understanding of function
Fair understanding of function
Weak understanding of function.
The ranges in Table 4 were determined by combining the ranges from Tables 2
and 3. The upper range combines the ranges for a student who is at the process level
(score of 90-96 on Questionnaire 1) from Table 2 with the range for having a strong
definition of function (score of 21-24 on Questionnaire 2) from Table 3. The lowest range
is a combination of ranges for students who are at the pre-action level (score of 24-37 on
Questionnaire 1) with the range for a weak definition (score of 8-12 on Questionnaire 2)
of function. The 75-111 range combines the range for students in the pre-process level
(score of 60-89 on Questionnaire 1) and the range for having an average definition of
31
function (score of 13-20 on Questionnaire 2). The range of scores between the average
understanding and weak understanding (50-72) were assigned to the fair understanding.
I used Excel to perform a linear regression between the composite score and the
student’s semester exam grade to determine whether there was a correlation between
student understanding of function and performance in calculus. I then performed linear
regressions by course (AP Calculus versus Introduction to Calculus), using the composite
score and the scores on each questionnaire as independent variables. This allowed me to
answer question 1: Is there a correlation between student understanding of function and
performance in calculus? To answer question 2: What patterns in student understanding
of function are related (or unrelated) to student performance in calculus, an Excel
spreadsheet was created. First, each student composite score was aligned with the
student’s first semester cumulative Calculus exam grade. I then identified patterns in the
responses through an item-by-item analysis, a total questionnaire score analysis and
finally a composite score analysis. Through this comparison, it was noted whether there
was a pattern in responses that relates to student scores on the two questionnaires and
their level of success in the introductory calculus course. All data were stored in an Excel
spreadsheet so that the analysis was efficiently and accurately performed.
32
CHAPTER FOUR: RESULTS
In this chapter, I report the results of my analysis of students’ responses to the two
questionnaires, characteristics of the composite score groups, students’ exam scores, and
the regression analysis.
Questionnaire 1
Questionnaire 1 was used to determine at what level a student entering a first-year
calculus course was able to work with functions. The mean score for this questionnaire
was 60.8. This score is at the lower range of the pre-process ability with functions based
on the rubric for this questionnaire. It was found that of the 116 students who participated
in the study, one student performed at the process level, 61 students performed at the pre-
process level, 52 students performed at the action level and 2 students performed at the
pre-action level.
To determine whether there were any areas where student performance tended to
be stronger (or weaker), a table was created to show the percentage of students that were
successful on items in relation to each of the PCA taxonomy categories. The table below
(Table 5) shows the percentage of students who scored an average score greater than 2.5
on all items in the section of the taxonomy as well as the percentage of students who
scored an average of 2.5 or less on all items in the section of the taxonomy. The score of
2.5 was chosen as it was a convenient division point on an assessment that produced
scores between one and four points. Looking at Table 5, the row that corresponds to
33
items related to R1 (View function as a process) shows that 48% of the students scored an
average of more than 2.5 points on all the items related to this category and 52% of the
students earned an average score of 2.5 or less on these items.
Table 5: Percentage of students scoring an average greater than 2.5 or less than or equal to 2.5 on all items corresponding to sections of the PCA taxonomy
Percentage of
Average Scores > 2.5
Percentage of Average scores
≤ 2.5
PCA Taxonomy category
R1 (Function as a process) 48% 52%
R2 (Covariational reasoning) 54% 46%
R3 (Proportional reasoning) 35% 65%
C1 (Evaluate and interpret functions) 61% 39%
C2 (Represent function situations) 47% 53%
C3 (Perform function operations) 50% 50%
C4 (Inverse functions) 47% 53%
C5 (Interpret and represent function behaviors) 50% 50%
C6 (Rate of Change) 53% 47%
As a whole group, student performance was roughly equally split between the
groupings of scores that had an average greater than 2.5 and scores with an average less
than or equal to 2.5 on all categories of the taxonomy with the exception of categories R3
and C1. Category R3 was aimed at measuring the students’ abilities to engage in
proportional reasoning. The larger percentage of students had average scores less than or
equal to 2.5 in this category. Category C1 measured student ability to evaluate and
interpret function information given the function’s formula, graph and table. In this
34
category students performed well with 61% of students earning an average score greater
than 2.5 on all items in C1.
The following table (Table 6) shows student performance for each item on
Questionnaire 1. The first column shows the item number along with the appropriate
taxonomy category. The next four columns show the percentage of students that earned
each of the respective scores based on the rubric used in scoring the questionnaire.
Table 6: Item breakdown of Questionnaire 1
Item (taxonomy category) % of 4s % of 3s % of 2s % of 1s
1 (R1, C1, C3, C5) 58% 9% 8% 25%
2 (R1, C1, C3, C4) 47% 1% 24% 28%
3 (R3, C2) 46% 16% 10% 28%
4 (R2, C2) 34% 4% 30% 31%
5a (R1, C1, C3, C4) 43% 3% 18% 35%
5b (R1, C1, C3, C4) 46% 14% 4% 36%
6 (R1, C5) 38% 24% 17% 22%
7 (R1, R2, C1, C5, C6) 5% 28% 47% 20%
8 (R1, C1, C3, C4) 28% 9% 30% 33%
9 (R2, R3, C2, C5) 12% 18% 31% 39%
10 (R2, C5, C6) 6% 2% 64% 29%
11a (R1,C1, C3, C4) 32% 26% 12% 30%
11b (R1, C1, C3, C4) 11% 10% 26% 52%
12 (R1, C4) 12% 2% 23% 62%
13 (R2, C1, C2, C6) 42% 42% 4% 12%
14 (R1, C3, C4) 73% 3% 1% 23%
15 (R2, C2, C6) 17% 7% 50% 26%
16 (R2, C1) 64% 11% 5% 20%
17 (R2, C1, C5, C6) 55% 36% 1% 8%
18 (R1, R2, C3) 6% 18% 51% 25%
35
19 (R1, C1, C5) 6% 30% 18% 45%
20 (R2, C1, C5) 18% 17% 56% 9%
21 (R2, C1, C6) 65% 20% 1% 14%
22 (R2, C1, C5) 43% 15% 10% 32%
Items on this questionnaire were determined to be items on which students
showed a strong performance if the overall percentage of a score of 4 was greater than
60%. Therefore, items 14, 16 and 21 would be items on which students showed strong
performance. Item 14 had students evaluate the composition of two functions given the
function rules. Items 16 and 21 were items in taxonomy categories R2 and C1. These
items both had students interpreting the values of a function for particular input values.
Item 1 was not included as a strong performance item, first because the percentage of
scores of 4 did not exceed 60%. The item asked students to evaluate a given function at
an input of (x + a). Although students who scored a 2, 3, and 4 all showed the ability to
correctly begin the process of evaluating f(x + a), students who earned the lower scores
either did not simplify the expression or made algebraic or arithmetic mistakes in the
simplification process. Students who did not simplify were counted as earning a score of
4, but after discussion with the co-primary investigator, it was determined that we could
not determine whether the student had the ability to simplify the expression correctly.
Therefore, there is a possibility that the number of 4’s could be inflated. Finally, items 13
and 17 were not included as strong performance items, as a score of 3, based on the
rubric, would not necessarily show a strong ability to interpret the rate of change of a
function, which was the aim of both items. Both items allowed students the ability to
choose one of two possible answers counted as a score of 3 of the five total possible
36
choices. What this means is that a student had a 3 out of 5 chance of picking a response
and earning a score of either 4 or 3 showing a relative high ability with function.
Items 10 and 12 were items on which students showed weak performance. The
performance on these items was considered weak as more than 80% of the students
involved in the study received a score of 1 or 2 on these items. Item 10 had students
calculate the average velocity of a car over a period of time given the position function of
the car. Students primarily responded to this item by calculating the position of the car at
one of the given time values (score of 2) or were not able to respond to the item at all
(score of 1). Item 12, had students find the inverse function of a function given as a rule.
For this item, students either did not produce a function at all or students produced a
function that was the reciprocal of the given rule. Item 18 was not included as an item
with weak performance as it was determined that many of the students misinterpreted the
question.
Additionally, items 10 and 16 (both in category R2) and items 12 and 14 (both in
category R1 and C4) showed the trend that one of the items was considered an item on
which the students showed strong performance and the second item students showed
weak performance. For items 12 and 14, recall item 12 was considered as a weak
performance item while item 14 was considered a strong performance item. Although
they were both measuring abilities in the same taxonomy category, the amount of success
shown on the item could have been due to familiarity with the notation. For items 10 and
16, students were demonstrating their ability to apply covariational reasoning to interpret
function behavior. That is how does a change in one variable affect changes in the other
variable of an equation. In both items students were presented with an equation. Item 16
37
had students directly using the equation to interpret its behavior. While for item 10,
students had to understand how to use the given position function to find information
about velocity. The indirect use of the given equation and also unfamiliarity with the
measured concept was a possible cause of the poor performance on item 10.
Finally, items 14 and 16 had a very large percentage with a score of 4 as well as a
large percentage with a score of 1, with less than 20% of the students with scores of 2 or
3. This shows that students either demonstrated complete ability to answer the item
correctly, or demonstrated very little if any ability with the item. Students who scored a 4
on item 14 showed the ability to produce the appropriate composition of two functions
either by simply producing the correct answer or by showing the process that they used to
produce the correct solution. The students earning a score of 1 either were unable to
produce a solution or produced some form of product of functions, for example some
students wrote g(h(2)) = g(x) ⋅h(x)⋅2. A score of 4 was earned on item 16 if students
correctly chose the option that describes the behavior of a rational function.
Questionnaire 2
Questionnaire 2 was intended to measure a student’s ability to apply the definition
of function upon entering a first-year calculus course. Based on the summative score on
the questionnaire, each student was assigned a strength of ability to apply the definition
ranking: strong, average, or weak. The mean score for this questionnaire was 12.9. This
score is just below the low end of the range of an average ability to apply the definition of
function. Of the 116 students involved in the study, 1 student entered calculus with a
strong ability to apply the definition, 59 students entered with an average ability to apply
38
the definition and 56 students entered calculus with a weak ability to apply the definition
of function. The table below (Table 7) shows the breakdown of the percentage of students
who scored each of the possible scores of 3, 2, or 1 with a score of 3 being the highest on
each item of Questionnaire 2.
Table 7: Item breakdown of Questionnaire 2
Item % of 3s % of 2s % of 1s
1 7% 28% 65%
2 23% 53% 24%
3 6% 17% 78%
4 20% 35% 45%
5 57% 14% 29%
6 9% 22% 70%
7 13% 30% 57%
8 2% 35% 62%
As a group, the students demonstrated a lack of ability to explain their reasoning
as to how they determined whether the representations in items 1 through 5 were or were
not functions. Many students used reasons such as “passes the vertical line test” to justify
their conclusion. With more than 70% of the students scoring a 1, items 3 and 6 were
items on which the students showed weak performance. Item 3 asked students to
determine whether a scenario described in words was a function and then justify their
reasoning. Item 6 asked students to create a piece-wise function that satisfied particular
parameters. Although item 1 had a high percentage of 1’s, I felt that students did not pay
close attention to the “±” in the relation . Therefore, I did not judge this item
as one on which the students performed poorly. Because only 2% of the students
xxf ±=)(
39
involved in the study scored a 3 on item 8, it can be concluded that students were not able
to state a precise definition of function.
The table below (Table 8) shows the results of the two questionnaires and the
number of students in each category.
Table 8: Student results on the two questionnaires
Questionnaire 1 Questionnaire 2
Total Students (N) 116 Total Students (N) 116
Process 1 Strong Definition 1
Pre-Process 61 Average Definition 59
Action 52 Weak Definition 56
Pre-Action 2
From the table it can be concluded that the participants of this study entered
calculus performing primarily at the pre-process or action level and with a primarily
average or weak ability to apply the definition of function. The student who performed at
the process level was not the same student with the strong ability to apply the definition
of function.
Composite Group Results
Once both questionnaires were administered, each student was assigned a
composite score. This composite score separated each student into one of three
subgroups: average, fair, or weak. Each subgroup was based on a student’s understanding
of function as measured by the student’s composite score on the two questionnaires. A
student’s composite score on the two questionnaires was used to determine the level of
understanding of function for each student. As discussed in the literature, a student’s
40
ability to work with functions as well as their ability to apply the definition of function
may determine how well they understand functions.
After each questionnaire was analyzed separately, the composite scores were then
analyzed. Students were placed into groups and I tried to determine if there was a
particular characteristic of each group. Table 9 shows the percentage (number) of
students that scored in each of the composite categories.
Table 9: Composite score category percentages (based on scores on two questionnaires)
Total N = 116
Strong
(111-120 Composite
Score)
Average
(73-110 composite
score)
Fair
(50-72 composite
score)
Weak
(32-49 composite
Score)
Percentage (number) in each composite category
0% (0) 52.6% (61) 44% (51) 3.4% (4)
41
Figure 3: Graph of Student Results of Questionnaire 1 vs. Questionnaire 2
Figure 3 above is a scatter plot of student scores on the two questionnaires. The
horizontal axis is the range of scores for Questionnaire 2 and the vertical axis is the range
of scores for Questionnaire 1. The student scores within each subgroup are represented by
a different shape in the plot (weak understanding group are triangles, fair understanding
group are squares, and average understanding group are rhombi). The horizontal lines
show the upper value of the range for each of the levels from Questionnaire 1 (Pre-action,
action, pre-process, and process). From the graph, one could observe that a student
scoring a 10 on Questionnaire 2 scored in the range of 34 to 73 on Questionnaire 1. A
student scoring a 10 on Questionnaire 2 could be considered as have a pre-action, action
or pre-process ability to work with function as the points associated with a Questionnaire
2 score of 10 fall into each of the Questionnaire 1 ranges in the graph.
Pre-Action Range (24-37)
Action Range (38-59)
Pre-Process Range (60-89)
Process Range (90-96)
24
34
44
54
64
74
84
94
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Que
stio
nnai
re 1
Sco
re
Questionnaire 2 Score
Questionnaire 1 vs. Questionnaire 2 Scores
42
The table below (Table 10) shows the performance of each subgroup on
Questionnaire 1 items that are included in the taxonomy categories. The percentage of
students of each subgroup that are considered successful (an average score greater than
2.5 on all items in the category) as well as the percentage of students who are considered
not successful (average score of 2.5 or lower on all items in the category are listed for
each subgroup in Table 10. For example the first two cells in the row for R1 show that
85% of the students of the average understanding subgroup showed success on the items
in the category, while 15% of the students in the average understanding subgroup did not
show success with items in the category.
43
Table 10: Subgroup performance on Questionnaire 1 taxonomy categories
Student Group Based on Composite Questionnaire Score
Average (N=61) Fair (N=51) Weak (N=4)
Taxonomy Category
% of students with average score >2.5
% of students with average score ≤ 2.5
% of students with average score >2.5
% of students with average score ≤ 2.5
% of students with average score >2.5
% of students with average score ≤ 2.5
R1 85% 15% 10% 90% 0% 100%
R2 77% 23% 33% 67% 0% 100%
R3 51% 49% 20% 80% 0% 100%
C1 93% 7% 29% 71% 0% 100%
C2 67% 33% 27% 73% 0% 100%
C3 84% 16% 16% 84% 0% 100%
C4 80% 20% 12% 88% 0% 100%
C5 75% 25% 25% 75% 0% 100%
C6 84% 16% 41% 59% 0% 100%
The Average Understanding Group
A student in this study was placed in the average understanding subgroup if his or
her composite score fell in the range of 75 and 111 on the two questionnaires. Of the 116
students involved in the study, it was determined that 61 were in this subgroup.
Results from Questionnaire 1 determined that, in this subgroup, one student had
the ability to work with functions at the process level, 56 students worked with functions
at the pre-process level and four students worked with functions at the action level.
Therefore, students in this subgroup primarily have the ability to work with functions at
44
the pre-process level. The results of student performance in this subgroup in each of the
PCA taxonomy categories are shown in the table above.
A student in the average understanding subgroup had the ability to answer the
majority of the items in each category with a good level of success. The values in Table
10 show the percentage of students who answered questions in each category correctly
with an average score of more than 2.5 for all the items in that category. Much like the
group of students in this study as a whole, the average subgroup performed in a similar
manner on PCA taxonomy categories R3 (engage in proportional reasoning) and C1
(evaluate and interpret function information given the function’s formula, graph and
table). Of the students in the average subgroup, 93% were able to score an average of
more than 2.5 on all the items in category C1. This was this subgroups strongest category.
Although category R3 had 51% of the students show the ability to score an average score
of more than 2.5 on items in this category, the percentage was the lowest of all the
categories for this subgroup.
A student in the average understanding subgroup entered calculus primarily with
an average ability to apply the definition of function. Of the 61 students in this subgroup,
one student scored a strong ability to apply the definition of function on Questionnaire 2.
Forty-seven students scored an average ability to apply the definition of function while
13 students scored at the weak ability level. The students in the average subgroup
demonstrated the ability to correctly answer (an average group score greater than 2) two
items on questionnaire two. These two items included interpreting a table and an input-
output mapping diagram as a function (items 2 and 5 on Questionnaire 2).
45
The Fair Understanding Group
A student with a fair understanding of function scored a composite score between
50 and 72 on the two questionnaires. Fifty-one of the 116 students involved in the study
fell into this subgroup. Of these students 5 worked with functions at the pre-process level
while 46 of the students worked with functions at the action level. The group’s
performance by PCA taxonomy category is in Table 10.
To discuss any possible trends in student performance on the two questionnaires, I
will again focus on this subgroup’s performance in categories R3 and C1 of the PCA
taxonomy. Students of this subgroup had the third highest percentage of students show
the ability to score over an average of 2.5 points on all the items in category C1.
Although there was a greater percentage of students who scored less than 2.5, this
remains consistent with the whole group as a category on which the groups are
performing well, the percentage of students in the subgroup that scored an average score
of more than 2.5 point on all items in the category was among the greatest. Similarly,
category R3 had the same performance in this subgroup as it had in the whole group. It
was considered as a category on which the students showed poor performance since 80%
of the fair subgroup scored an average score less than 2.5 on all items in this category. In
addition, students in this subgroup showed poor performance in category R1 (View
function as a process) as only 10% of the students in the subgroup were successful with
items in this category.
Results of Questionnaire 2 showed that students in this subgroup primarily had a
weak ability to apply the definition of function. Twelve students began the year with an
average ability to apply the definition of function while 39 were measured as having a
46
weak ability to apply the definition of function based on results of Questionnaire 2. Here
again students performed the best on items 2 and 5. The average score of the entire group
was the highest on these two items. The average scores on the items were 1.8 and 2 for
items 2 and 5 respectively.
The Weak Understanding Group
A student was placed into the weak understanding subgroup if the composite
score on the two questionnaires was between 32 and 49. This subgroup contained four
students. Of the four students, two of the students demonstrated the ability to work with
functions at the action level while the remaining two students performed at the pre-action
level. All four students were measured as having a weak ability to apply the definition of
function.
No student in this subgroup was able to answer all the questions in any of the
PCA taxonomy categories with an average of greater than 2.5. The students in this
subgroup showed the greatest performance on items 1, 6, 13, and 21 of Questionnaire 1
with average scores of 2.25, 2.5, 3, and 2.5 respectively. Items 1, 13, and 21 are all items
in taxonomy category C1, the category on which the whole group is showing the greatest
performance. Because of the overall performance of the weak understanding subgroup,
the trend with category R3 was maintained in this subgroup as one in which the students
demonstrated low performance.
By analyzing answers on Questionnaire 2 for this subgroup, it should be noted
that students received primarily a score of one (the lowest possible score) on the items.
One student received a score of 3 on item 5 (the input-output mapping diagram). Also,
47
only one student scored a 2 on item 2. Item 2 presented students with a relation in the
form of a table and asked them to determine whether the relation represented a function.
Determining Success in Calculus
To determine which students were successful, recall that I used the students’
cumulative exam scores for the first semester. This means that the scores earned on all
exams in the first semester were averaged. The average of all the scores was used to
determine whether or not the student was successful. Table 11 shows the percentage of
students that were considered as successful in the two courses. For example, the second
row and second column of Table 11 shows the percentage of successful students (36%)
along with the number of successful students (22) who were enrolled in AP Calculus AB.
Table 11: Student breakdown by calculus course.
Calculus Class % Successful % Not successful
AP Calculus AB (N=60) 36% (N=22) 64% (N=38)
Introduction to Calculus (N=56) 30% (N=17) 70% (N=39)
Although one might consider that any grade higher than an 80% would be
successful in high school calculus, with these students being some of the higher achieving
students, I felt that higher percentages should be required in order for the student to be
considered successful. For reference, the mean exam score for the AP Calculus AB
students was 84.7%, while the mean exam score for the Introduction to Calculus students
was 80.7%. Table 12 contains other basic statistics associated with the exam scores for
each course. As could be seen in Table 12, the values in the columns labeled AP Calculus
and Introduction to Calculus are somewhat similar. However, when comparing the
48
standard deviation of the exam scores for the two courses, notice that the scores of
Introduction to Calculus have a greater distribution than those of AP Calculus.
Table 12: Basic Statistics on Exam Scores
Statistic AP Calculus Introduction to Calculus
Mean 84.7 80.7
Standard Deviation 9.4 13.8
Min (Course Exam %) 57.7 50.6
Max (Course Exam %) 101.4 101.2
N 60 56
The figures below show the distribution of exam score percentages for the two classes
involved in this study, AP Calculus AB (Figure 3) and Introduction to Calculus (Figure
4).
Figure 4: Distribution of exam scores for students enrolled in AP Calculus AB
0
5
10
15
20
25
30
>88 78-87 68-77 <67
Num
ber o
f Stu
dent
s
Average exam score percentage
AP Calculus Exam Score Distribution
49
Figure 5: Distribution of exam scores for students enrolled in Introduction to
Calculus
Once students were determined as being in either the success group or the not
successful group, I then tried to determine if there were any trends among the scores in
each of the groups. The results of this initial analysis are shown in Table 13 below. The
percentages in the table describe the percent of students in that category that were
considered to have an average, fair, or weak understanding of function based on results of
the two questionnaires. For example, looking at the first row and third column of Table
13, one would see that 81.8% (18 students), enrolled in AP Calculus AB, who were
considered successful in the course scored as having an average understanding of
function based on results of the two questionnaires. The other 18.2% (4 students) of the
successful AP students demonstrated a fair understanding of function.
0
2
4
6
8
10
12
14
16
18
>=91 81-90 71-80 61-70 <61
Num
ber o
f stu
dent
s
Average exam score percentage
Introduction to Calculus Exam Score Distribution
50
Table 13: Successful and not successful student percentages at each level of understanding of function.
% average % fair % weak A
P C
alcu
lus A
B
(N=6
0)
Successful (exam average > 88%) N=22 81.8%
(N=18)
18.2%
(N=4)
0%
(N=0)
Not successful (exam average ≤ 88%) N=38 73.7%
(N=28)
26.3%
(N=10)
0%
(N=0)
Intro
duct
ion
to
Cal
culu
s (N
=56)
Successful (exam average > 91%) N=17 64.7%
(N=11)
29.4%
(N=5)
5.9%
(N=1)
Not successful (exam average ≤ 91%) N= 39 10.3%
(N=4)
82.1%
(N=32)
7.7%
(N=3)
It is fairly surprising that in the AP Calculus AB class that the percentage of
students that were considered as having an average versus fair understanding of function
were fairly close in both the successful and not successful categories. In other words, the
rate of success for a student enrolled in AP Calculus was not determined by the level of
understanding the student had with function. However, of the 28 students enrolled in AP
Calculus AB with an average understanding of function that were not considered
successful in the course, sixteen of these students scored an average exam score between
78% and 87% during the first semester.
The Introduction to Calculus class was much more telling. The majority of the
students that were considered successful started the year with an average understanding
of function. There was also a fair percentage of students that were considered successful
that started the year with a fair understanding of function. The group of students who
51
were not considered to be successful in the Introduction to Calculus course had a very
heavy percentage of students that had a fair understanding of function at the beginning of
the course. Very few students who were considered not successful in Introduction to
Calculus had an average understanding of function. A student’s rate of success in
Introduction to Calculus appears to be more closely related to their measured
understanding of function than it was in AP Calculus AB. That is a student with an
average understanding of function had a greater rate of success in the course over a
student with a fair or weak understanding of function.
Recall that I assigned each student a semester exam grade based on the
cumulative total of all the exams taken the first semester of the respective introductory
calculus course. I used Excel to perform a linear regression between the composite score
and the student’s semester exam grade to determine whether there was a correlation
between student understanding of function and performance in calculus. I found that
student exam scores can be predicted by student composite scores with following
equation: Exam Score=56.5+0.356*Composite Score (R=0.426; S=10.77; p=0).
I also performed linear regressions by course (AP Calculus versus Introduction to
Calculus), using the composite score and the scores on each questionnaire as independent
variables. These results are illustrated in Table 14. The column labeled R (Exam score vs.
Composite Score) shows the correlation for each course involved in the study when
taking the composite score as the independent variable. The asterisk (*) indicates that the
exam score could be predicted by the corresponding independent variable for students in
a particular course at the p < 0.05 level. The p-value for each regression score is listed in
each cell in parentheses.
52
Table 14: Correlation between average exam scores and questionnaire scores by course.
R (Exam score vs. Composite score)
R (Exam score vs. Questionnaire 1 score)
R (Exam score vs. Questionnaire 2 score)
AP Calculus AB .254
(p=0.050)
.266*
(p=0.0398)
.089
(p=0.499)
Introduction to Calculus
.537*
(p=0.00002)
.512*
(p=0.00005)
.361*
(p=0.0063)
The scores for Questionnaire 1 show statistical significance at the p < 0.05 level
for both the AP Calculus AB and the Introduction to Calculus courses. That means that
one could reasonably predict the exam score for a student knowing the score a student
earned on Questionnaire 1. This result was also true for the composite score and the
Questionnaire 2 score of the Introduction to Calculus student scores. The predictability of
the results is however limited as the standard error in all cases is quite large.
The composite scores and the Questionnaire 2 scores did not show statistical
significance at the p < 0.05 level as predictors of exam scores for the students enrolled in
AP Calculus AB. That means, that knowing either a student’s composite score or
Questionnaire 2 score, one would not be able to determine that student’s semester exam
grade in the course. The linear regression models for each of the statistically significant
results are shown in Table 15 below. Residual plots were checked for each to be sure
there were no patterns in the residuals.
53
Table 15: Linear Regression models for statistically significant regression data.
Course (Independent Variable)
Regression Model Standard Error
AP Calculus AB (Questionnaire 1 Score)
Exam %= 68.5802 + 0.2392 * Q1 Score
9.12
Introduction to Calculus (Composite Score)
Exam %= 38.2339 + 0.6484 * Comp. Score
11.76
Introduction to Calculus (Questionnaire 1 Score)
Exam % = 43.1584 + 0.6972 * Q1 Score
11.97
Introduction to Calculus (Questionnaire 2 Score)
Exam % = 58.5322 + 1.9003 * Q2 Score
13.00
54
CHAPTER FIVE: DISCUSSION
Students entering calculus at the high school level would, in most people’s eyes,
already be considered successful. However, in this study, I aimed to determine whether
understanding of function plays a role in student success as well as if there were any
patterns in student understanding that contributed to the student’s success in calculus. In
this chapter, I use the results of my analysis to answer my research questions and discuss
some possible limitations of this study.
Is There a Correlation Between Student Understanding of Function and
Performance in Calculus?
The data indicate that student performance on Questionnaire 1 serves as a
reasonable predictor of student success in calculus in a limited way. The results showed
that exam scores could be predicted based on scores from Questionnaire 1 for both AP
Calculus AB as well as Introduction to Calculus. In other words, a student’s ability to
work with functions would translate into roughly the appropriate level of success in
calculus (working with functions at the process level would translate into a high rate of
success while working with functions at the pre-action levels would translate into a lower
rate of success). Questionnaire 2 can serve as a limited predictor for students enrolled in
Introduction to Calculus, however, not for AP Calculus AB. While the composite score
for the entire group appeared to be a predictor of exam score, this result did not carry
through when examined by course. That is, the composite score could be used to predict
55
whether a student entering Introduction to Calculus could be successful, however, the
composite score is not a good predictor of exam score (or success) for a student entering
AP Calculus AB.
While analyzing the results of Questionnaire 1, I discovered interesting trends in
performance for the students that were involved in the study. From this questionnaire, I
found that the students entered calculus with the ability to work with functions at either
the action or pre-process level of the APOS framework. The results of Questionnaire 2
showed that students entered calculus with difficulty defining function in line with the
Dirichlet-Bourbaki definition. The participants of the study began the calculus course
with either an average or weak ability to apply the definition of function.
The composite scores showed that students entered their respective first-year
calculus courses with either an average or a fair understanding of function based on
composite scores. The most significant finding related to success is that students can be
successful in a first year calculus course without demonstrating a process level
understanding of function at the beginning of the course. Further, it is possible for
students to be successful in calculus with an average or fair understanding of function
although generally much less likely for students with a fair understanding compared to
those with an average understanding of function.
For students enrolled in the AP Calculus AB course, the percentage of students
with an average understanding of function compared to those with a fair understanding
was almost equal in the successful versus not successful categories. However, these
students do all come from Honors Pre-Calculus which is taught by one teacher at the
school where this study was conducted. Therefore, material prior to calculus should have
56
been presented to the students in similar manners. Therefore student performance should
be somewhat more consistent for the students entering AP Calculus AB.
The difference between the students entering Introduction to Calculus as opposed
to those entering AP Calculus AB is that the students entering Introduction to Calculus
have the possibility of either taking Honors Pre-Calculus or Math Analysis prior to the
calculus course. Because of this, students frequently enter the class with varying degrees
of mathematical understanding and ability. These two courses were also taught by three
different teachers the year prior to this study. As mentioned earlier, a student who was
enrolled in Introduction to Calculus and was measured as having an average
understanding of function, had a high rate of success in the course. Whereas, a high
percentage of students enrolled in the same class with a fair or weak understanding of
function were not considered successful in the course.
What Patterns in Student Understanding of Function Are Related (or Unrelated) to
Student Performance in Calculus?
The results of the two questionnaires showed that students entered their respective
first-year calculus courses with a varied understanding of function. As a group, the
students showed the ability to correctly solve problems with which they should have been
familiar. These items included ones where the student was given the algebraic rule and
asked to calculate the value of the function. Here students are working with functions at
the action view of function. When presented with a problem in context, or in a form with
which the student is not familiar, these were items where students tended to demonstrate
weaker performance. Overall, students demonstrated difficulty stating the definition of
57
function precisely, which likely contributed to the difficulty students had using the
definition to justify decisions
Students had the ability to answer items on Questionnaire 1 that were familiar
from previous mathematics courses. The PCA taxonomy category that included items
which are typically presented in a standard mathematics text are in category C1, the
category which, recall from the results chapter, was the category where the students
showed a strong performance. These items involved giving the students the function as a
rule and finding the value of the function at a particular input. Items 1 and 14 were these
kinds of items. However, if students were presented with a function as a table or graph
and asked to perform the same task, the percentage of 3s and 4s was not as high as when
the students were provided with the rule. It is interesting to note that 75% of the students
involved in the study were able to start correctly when determining f(x + a) on item
number 1 (students with scores of 2, 3, or 4). However, when we began the study of the
derivative, this concept seemed difficult to the students. This could be because the idea is
combined with many other concepts and not just simply evaluating a function for a
particular input.
Items 16, 17, 21, and 22 are items that are covered near the end of typical courses
taken prior to a student entering calculus. Recall from the results chapter, Table 6 showed
a large percentage of students showing the ability to score either a 3 or 4 on three of the
four items. These items should be the most familiar since they were introduced at the end
of the course typically taken prior to calculus. Many of the other items on which the
students showed success are repeatedly covered throughout the high school mathematics
curriculum. Therefore, it is not a surprise that students were successful with these items.
58
Two items on which the students showed the ability to earn scores of 3s and 4s,
items 13 and 17, were multiple-choice items. Unlike the “choose all that apply” items, 20,
21, and 22, it was difficult to tell the ability a student had with function from these items
as I felt that some students could have guessed the correct choice and earned a 4 or 3 on
that item without having true ability to work with functions in the way intended by the
item.
Item 5 on Questionnaire 2 showed a fair amount of success by students as a
majority of students (57%), earned a score of 3. This item asked students to determine
whether an arrow diagram mapping of domain elements to range elements was a
function. This is typically how functions are presented to students in courses prior to
calculus. Therefore, I feel that the students’ familiarity with the form of the item was a
primary cause for their success.
When students were asked to work with functions in an unfamiliar way, the
performance on the items was not as high as when they were provided with the function
as an algebraic rule, even for items within the same taxonomy category. The students’
difficulty came when asked to use functions, whether in equation, graphical, or tabular
form, in ways that were unfamiliar to them. For example, item 2 asked students to find
the input of a function that produced a particular output given the graph of a function. A
second example, students were considered very successful on item 14, where they were
asked to calculate the composition of two functions for a particular input given each
function as an algebraic rule. However, when asked to perform a similar task on item 11a
given the table, the level of success decreased. This mirrors much discussion in the
59
literature that students have difficulty connecting the different forms of function: rule,
table, or graph (Thompson, 1994; Davis, 2007; Carlson, et al., 2010; Akkus et al., 2008).
Next, students demonstrated that they had difficulty calculating inverse functions.
Item 11b presented students with a table and asked students to identify the value of the
inverse function at a given input. Here students either did not provide an answer at all,
provided an answer, which showed no connection to the problem, or simply found the
function value rather than the inverse function value. Item 12 was another problem asking
students to calculate the inverse of a given function. This time however, students were
provided with the rule and asked to find the inverse function as a rule. On this item again,
the incorrect responses either involved no solution at all or students interpreted the
notation f -1 as the reciprocal of the function f.
Also, items 7 and 10 proved as items with which students had difficulty. Items 7
and 10 had students connecting position and velocity information. Item 7 presented
students with the graph of the speed of two cars with respect to time. The students were
asked to identify the position of one car relative to the other at a particular time. Students
used the fact that the two graphs intersected to conclude that the cars were at the same
position at the time in question. Here students disregarded the information that they were
being presented with in the graph. Much like what has been mentioned in the literature,
for these students, the shape of the graph models the “trip” of each car (Monk, 1992,
Oehrtman et al., 2008). On item 10, students were given the position of a car relative to
time as an algebraic rule. Here students were asked to calculate the average velocity over
a given period of time. The common solutions that students produced were, first the
position of the car at either or both of the given times. Here students simply evaluated the
60
given function at the given values, relying on a concept that was familiar from previous
mathematics courses. Second, students found the sum of the position at the beginning of
the time interval and the position at the end of the time interval and divided the sum by
two. In other words, some students calculated the average of the two positions. It was not
surprising to me that these two items proved as a challenge to students, as these are
concepts that are covered in calculus and physics. Many students entering calculus are
concurrently enrolled in physics.
However, much like Questionnaire 1, items on Questionnaire 2 that were not
familiar to students, such as functions which were not continuous, functions written as a
verbal statement, and piece-wise functions all gave the students difficulty. This again is
consistent with the literature (Dubinsky & Wilson, 2013; Sierpinska, 1992; Clement,
2001). Students also tended to use very simple explanations to explain their choices.
These included “the vertical line test” or continuity as how they determined whether a
relation was or was not a function. This was also mentioned in the literature (Dubinsky &
Wilson, 2013).
Limitations
Although some of the results of the study proved to be statistically significant,
there are some limitations that need to be discussed.
First, the administration of the questionnaires may not have been done in a way to
truly measure student ability. Being administered at the beginning of the school year,
there is a possibility that students may have simply forgotten some basic concepts on
which possibly a little review may have sparked their memory. Also, since the results of
the questionnaires were not connected to the students’ grades, some students may have
61
not taken the process of answering the items as seriously as they would have if the results
were part of their grade.
Next, the results of some of the items on both questionnaires may be skewed. For
example, as discussed earlier, some of the items being multiple choice could have
encouraged guessing. Therefore, a student may have earned full points on the item
without truly having the ability to understand why he or she was making the choice they
did. Also, some items presented problems in grading. An example of this occurred on
item 18. Students were given a function S(m) which was an employee’s salary per month
after m months on the job. Students were asked to explain what the function S(m + 12)
represented. To earn full points, I was looking for solutions like S(m +12) as a horizontal
shift left 12 units of the graph of S(m) or the salary after m + 12 months on the job. Many
students here provided answers like a “raise”. As such, it was difficult to interpret what
this meant in the mind of a student. Therefore, the score received for an answer such as
this may not have correlated to the student’s true understanding. Therefore, it may be
beneficial in the future to conduct interviews on particular items.
Frequently, in a calculus class at the high school level, students are high achieving
and motivated students. Therefore, one may wonder whether the student’s success was
truly due to the fact that they have a strong understanding of function or whether their
motivation to perform at a high level determined their success. Also is it possible that
some of the students with a strong definition of function or at the process level may not
perform very well in calculus due to a lack of effort that they are putting into the course.
Finally, the school at which the research was conducted is in an upper-middle to upper
class area. Therefore, the question arises as to whether the same results would be
62
obtained in a more rural or lower income area. The purpose of the study was, in part, to
begin the process of discussing what effect level of understanding of functions plays on
success in upper level high school mathematics classes like calculus.
63
CHAPTER SIX: CONCLUSIONS
In this chapter, I share some implications of the study from the perspective of a
teacher and some ideas for future research.
Implications as a Teacher
The results of this study have confirmed beliefs that I had about the relationship
between how well a student understands functions and how well the student performs in
class. Based on the results of Questionnaire 1, students who have a higher ability to work
with functions based on the APOS framework tended to be more successful in their
respective first-year calculus course. The concept of function is introduced to students
beginning in the eighth grade. Students entering calculus should have the ability to work
with functions in a variety of ways given the exposure to the concept through their
studies. That being said, as teachers, we must present students with functions in a variety
of contexts and representations. We, the teachers, must also allow for our students to
make connections between ideas presented throughout the course. For example, when
making the connection between the equation of a function and its graph, we probably
need to get away from the “replace f(x) with y” mentality and, instead, help our students
understand the meaning of the f(x) notation. This would help students move away from
the action level of APOS and begin viewing functions at the process level. Supporting
students in understanding at all levels of the APOS framework and applying the
definition of function must also be an area of improvement in mathematics courses.
64
The results of my study provide an excellent opportunity to discuss with teachers
of all mathematics classes to the importance of introducing students to functions using a
variety of representations. The results of the study showed students had difficulty
interpreting graphs, tables and verbal phrases in the context of function. Therefore much
more emphasis needs to be placed on these kinds of problems in the course of a student’s
studies of mathematics. Also, getting away from exclusively using x as the input and y as
the output would be of benefit to students and building their confidence when working
with functions. Finally, to possibly assist in student understanding of function, as teachers
we need to ensure that all students are defining function in ways consistent with the
Dirichlet-Bourbaki definition of function. This study showed that students primarily used
the vertical line test or continuity to justify their reasoning as to why a relation was a
function. Taking a close look at how students work with and define functions will inform
my teaching so that I might provide better support for students enrolled in first-year
calculus courses in the future.
It is recommended that much more emphasis be placed on functions in classes
prior to calculus in order for all students entering calculus to be successful. This focus
must be on all representations of functions: equation, graphical, tabular, and description
of a contextual situation. Involving teachers in the discussion of concepts that students
struggle with in calculus would benefit not only the calculus teacher (and students) but
also the teachers (and students) of courses prior to calculus.
Future Research
As a researcher, it would be interesting to see whether putting weight on the
questionnaires would affect the results of the study. I feel that some students may not
65
have taken the questionnaires seriously and therefore the results were not a true measure
of their ability. It would also be interesting to conduct a similar study on students entering
calculus that have gone through curriculum that aligns with the Common Core State
Standards in Mathematics (CCSSM; Common Core State Standards Initiative, n.d.) I am
curious to know whether or not these students would show better ability with functions
than those that came from the standard track of study in mathematics.
It would also be nice to simply focus on a smaller group of students. This way
interviews could be conducted and I could more accurately discuss what the student
understands or has the ability to do. Many times in this study, I found myself trying to
read into the solutions that students provided and possibly misinterpreted what they were
trying to present to me.
There is a possibility that success on the exams that were part of the calculus
courses involved in this study did not heavily rely on a dynamic view of function. For
this reason, a follow-up study could be conducted to perform an analysis of the course
exams. An alignment between the APOS theory and the exams would reveal the level of
demand of function understanding required.
According to Carlson (1997), students must possess a dynamic view of function,
and they must view functions at least at the process level. The results of this study proved
otherwise. Again it is reasonable to wonder whether this is a case of students not taking
the questionnaires seriously or whether or not what they produced a true assessment of
their ability. Regardless, the results of this study do provide evidence that students with a
pre-process view and limited ability to apply the definition of function do have a chance
at success in a first-year calculus course.
66
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(1996). A framework for research and curriculum in undergraduate mathematics
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Akkus, R., Hand, B., & Seymour, J. (2008). Understanding students’ understanding of
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Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the
process conception of function. Educational Studies in Mathematics, 23, 247-285.
Carlson, M. (n.d.). A study of second semester calculus students’ function conceptions.
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Carlson, M., Oehrtman, M., & Engelke, N. (2010). The precalculus concept assessment: a
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Clement, L.L. (2001). What do students really know about functions. Connecting
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Cooney, T., Beckman, S., & Lloyd, G. (2010). Developing Essential Understanding of
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Council of Teachers of Mathematics.
67
Davis, J. D. (2007). Real world contexts, multiple representations, student-invented
terminology, and y-intercept. Mathematical Thinking and Learning , 9, 387-418.
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Dubinsky, E. & MacDonald, M. (2001). APOS: a constructivist theory of learning in
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Mathematics at University Level: An ICMI Study (pp. 273-280). Dordrecht:
Kluwer Academic Publishers.
Dubinsky, E. & Wilson, R. T. (2013). High schools students’ understanding of the
function concept. The Journal of Mathematical Behavior, 32, 83-101: doi:
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Edwards, B., Ward M. (2008). The role of definitions in mathematics and in
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Habre, S. & Abboud, M. (2006). Students’ conceptual understanding of a function and its
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Kinzel, T. A. (2006). Exploring process/object duality within students’ interpretations
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Lobato, J. (2008). When students don’t apply the knowledge you think they have, rethink
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the Connection: Research and Teaching in Undergraduate Mathematics
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255-274). Washington, DC: Mathematical Association of America.
Monk, S. (1992). Students’ understanding of a function given by a physical model. In G.
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Mathematical Association of America.
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Moore, T. (2012). What do calculus students learn after calculus. (Unpublished
Dissertation). Kansas State University, Manhatten, Kansas.
National Research Council. (1989). Everybody Counts.
Oehrtman, M., Carlson, M., Thompson, P.W. (2008). Foundational reasoning abilities
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https://mathed.asu.edu/instruments/PCA/.
69
Ronda, E.R. (2009). Growth points in students’ developing understanding of function in
equation form. Mathematics Education Research Journal, 21, 31-53.
Sajka, M. (2003). A secondary student’s understanding of the concept of function – a
case study. Educational Studies in Mathematics, 53, 229-254.
Sierpinska, A. (1992). On understanding the notion of function. In G. Harel & E.
Dubinsky (Eds.) The Concept of Function Aspects of Epistemology and
Pedagogy, MAA Notes Volume 25 (pp. 25-58). Washington, DC: Mathematical
Association of America.
Thompson, P.W. (1994). Students, functions, and the undergraduate curriculum. In E.
Dubinsky, A.H. Schoenfeld, & J. J. Kaput (Eds.), Research in Collegiate
Mathematics Education, 1 (Issues in Mathematics Education Vol. 4, pp. 21-44).
Providence, RI: American Mathematical Society.
Vinner, S. (1992). The function concept as a prototype for problems in mathematics
learning. In G. Harel & E. Dubinsky (Eds.) The Concept of Function Aspects of
Epistemology and Pedagogy, MAA Notes Volume 25 (pp. 195-213). Washington,
DC: Mathematical Association of America.
Vinner, S. & Dreyfus, T. (1989). Images and definitions for the concept of function.
Journal for Research in Mathematics Education, 20, 356-366.
70
APPENDIX A
Questionnaire 1: Modified PCA (Precalculus Assessment, n.d.)
71
Questionnaire 1
Measurement of Ability to Work with Functions
Name___________________________________
Answer each of the following to the best of your abilities. Show all work that lead to your conclusion or include a description of how you arrived at your conclusion.
1. Given the function, f, defined by , find .
2. Use the graph to solve for x.
423)( 2 −+= xxxf )( axf +
3)( −=xf
72
3. To the left are drawings of a wide and a narrow cylinder.
The cylinders have equally spaced marks on them. Water is poured into the wide cylinder up to the fourth mark (see A). This water rises to the sixth mark when poured into the narrow cylinder (see B). Both cylinders are emptied and water is poured into the narrow cylinder up to the 11th mark. How high would this water rise if it were poured into the empty wide cylinder?
4. Write a formula which defines the area of a square, A, in terms of its perimeter, P.
5.
a. Use the graphs of f and g above to find g(f (2)).
b. Evaluate f (2) – g (0) using the graphs of f and g above.
73
6. The model for the number of bacteria in a culture has been updated from to where t is measured in days. What implications can
you draw from this new model? (Choose one answer below) a. The final number of bacteria is three times as much of the initial value. b. The initial number of bacteria is 3. c. The number of bacteria triples every day. d. The growth rate of bacteria in the culture is 30% per day. e. None of these
7. The graph below shows the speed of two cars during a one-hour period. Assume the cars start at the same point and at the same time and are traveling in the same direction.
What is the relationship between the position of car A and car B at t = 1 hour?
ttP )2(7)( = ttP )3(7)( =
74
8.
Use the graphs of f and g above to find the values of x for which g(x) > f (x).
9.
A hose is used to fill an empty wading pool. The graph above shows volume (in gallons) in the pool as a function of time (in minutes). Define a formula for computing the time, t, as a function of the volume, v.
10. The distance, s (in feet) traveled by a car moving in a straight line is given by the function, , where t is measured in seconds. Find the average velocity, in feet per second, for the time period from t = 1 to t = 4.
ttts += 2)(
75
x f(x) g(x)
-2 0 5
-1 6 3
0 4 2
1 -1 1
2 3 -1
3 -2 0
11. a. Given the table above, determine f (g(3)).
b. Given the table above determine .
12. Given that f is defined by f (t) = 100t, define a formula for .
13.
The above graph represents the height of water as a function of volume as water is poured into a container. Which container is represented by this graph?
14. Given the function and , evaluate g(h(2)).
)1(1 −−g
1−f
13)( −= xxh 2)( xxg =
76
15. A ball is thrown into a lake, creating a circular ripple that travels outward at a speed of 5 cm per second. Express the area, A, of the circle in terms of the number of seconds, s, that have passed since the ball hit the lake.
16. The wildlife game commission poured 5 cans of fish (each can contained approximately 100 fish) into a farmer’s lake. The function N defined by
represents the approximate number of fish in the lake as a
function of time (in years). Which of the following best describes how the number of fish in the lake changes over time?
A. The number of fish gets larger each year, but does not exceed 500. B. The number of fish gets larger each year, but does not exceed 1200. C. The number of fish gets smaller each year, but does not get smaller than 500. D. The number of fish gets larger each year, but does not exceed 600. E. The number of fish gets smaller each year, but does not get smaller than 1200.
17. Using the graph below, explain the behavior of function f on the interval from x = 5 to x = 12. A. Increasing at an increasing rate. B. Increasing at a decreasing rate. C. Increasing at a constant rate. D. Decreasing at a decreasing rate. E. Decreasing at an increasing rate.
18. If S(m) represents the salary (per month) of an employee after m months on the job, what would the function R(m) = S(m + 12) represent?
15.05600)(
++
=tttN
77
19. What is the domain of the following function?
20. A baseball card increases its value according to the function, where b
gives the value of the card (in dollars) and t is the time (in years) since the card was
purchased. Which of the following describe what conveys about the situation?
I. The cards value increases by $5 every 2 years.
II. Every year the cards value is 2.5 times greater than the previous year.
III. The cards value increases by dollars every year.
(More than one choice may describe the situation. Choose all that apply)
21. A function f is defined by the following graph. Which of the following best describes the behavior of f ?
I. As the value of x increases, the value of f increases.
II. As the value of x increases, the value of f approaches 0.
III. As the value of x approaches 0, the value of f approaches 0.
(More than one choice may be correct. Choose all that apply.)
12)(
−+
=xxxf
10025)( += ttb
25
25
78
22. Which of the following best describes the function f defined by, ?
I. As the value of x gets very large, the value of f approaches 2.
II. As the value of x gets very large, the value of f increases.
III. As the value of x approaches 2, the value of f approaches 0.
(More than one choice may be correct. Choose all that apply.)
2)(
2
−=
xxxf
79
APPENDIX B
The Precalculus Assessment (PCA) Taxonomy Aligned With Questionnaire 1 Items
(Precalculus Assessment, n.d.)
80
The Precalculus Assessment (PCA) Taxonomy Aligned With Questionnaire 1 Items
Items in parentheses are items from Questionnaire 1 that require the specific ability
Reasoning Abilities R1 View function as a process.
• View a function’s formula, graph, and table as defining relationships that accept input and produces output.
(Items: 1, 2, 5a, 5b, 6, 7, 8, 11a, 11b, 12, 14, 18, 19)
R2 Apply covariational reasoning.
• Coordinate two varying quantities that change in tandem while attending to how the quantities change in relation to each other.
(Items: 4, 7, 9, 10, 13, 15, 16, 17, 18, 20, 21, 22)
R3 Engage in proportional reasoning.
• Vary the measures of two quantities and recognize that they are proportionally related when the measures of the two quantities are always in the same ratio.
• Recognize that when two quantities’ measures are always in a constant ratio then the measure of one is always the same multiple of the measure of the other.
(Items: 3, 9)
Conceptual and Analytic Abilities C1 Evaluate and interpret function information given the function’s formula, graph and table.
(Items: 1, 2, 5a, 5b, 7, 8, 11a, 11b, 13, 16, 17, 19, 20, 21, 22)
C2 Represent contextual function situations using algebraic notation.
• Identify, define and relate variable quantities as functional relationships.
(Items: 3, 4, 9, 13, 15)
81
C3 Perform function operations and interpret their meaning (table, graph, formula)
C3E Evaluate a function value and interpret its meaning. (Items: 2)
C1D Interpret domain restrictions inherent in the function. (Items: 19)
C3A Understand and use function arithmetic. (Items: 1, 5b, 8)
C3C Understand and use function composition. (Items: 5a, 11a, 14)
C3I Understand and use function inverse. (Items: 11b)
C3T Understand and use function translations. (Items: 18)
C4 Understand the meaning of an inverse function and how to reverse the function process (table, graph, formula)
C4E Solve equations that involve functional relationships and interpret their meaning.
(Items: 2, 5a, 5b, 11a, 14)
C4IN Solve inequalities that involve functional relationships and interpret their
meaning. (Items: 8)
C4IF Determine inverse functions and interpret their meaning. (Items: 11b, 12)
C5 Interpret and represent function behaviors for various function types.
C5L Linear (Items: 9, 20)
C5P Polynomial (Items: 1, 7, 10, 17)
C5R Rational (Items: 16, 19, 22)
C5E Exponential (Items: 6, 7)
C5L Logarithm (Items: 7)
C6 Interpret and represent rate of change information for a function (table, formula, graph)
C6D Understand and represent the meaning of the slope of a linear function as additive growth. (Items: 20)
C6C Interpret and represent how the input and output variables change in tandem.
(Items: 7, 13, 15, 16, 21)
C6A Determine and understand average rate-of-change. (Items: 10)
82
C6I Interpret and represent rate-of-change information on intervals of the domain.
(Items: 13, 17)
C6E Understand exponential functions and multiplicative growth. (Items: 7)
83
APPENDIX C
RUBRIC for Questionnaire 1
84
RUBRIC for Questionnaire 1
Item 1
4 3 2 1
Student correctly evaluates and simplifies f (x + a)
Evaluates f (x + a) with arithmetic errors.
Evaluates f (x + a) with some algebraic mistakes (ie. Distributive property)
Evaluate f (x) + a or other
Item 2
4 3 2 1
Produces -4 as the solution
Produces 8 as the solution. (understood concept but did not see the negative sign)
Produces -2 as the solution. (Reversed input output values)
Produces -3 as a solution or unable to produce a solution
Item 3
4 3 2 1
Correctly uses proportional reasoning to determine the solution (22/3)
Correct setup for proportional reasoning with minor arithmetic error (i.e. Multiplies by the wrong scale factor)
Approximates the solution to be between two values.
Uses additive reasoning. (ie. Since pouring the big into the little, went up 2 marks, little in big should go down 2 marks) or no solution given
85
Item 4
4 3 2 1
Produces the
formula: ,
Produces the formula:
,
Other formulas involving area and perimeter
Formula for either area or perimeter or unable to produce a formula.
Item 5a
4 3 2 1
Finds Finds
(Reversed composition) or
f(g(2)) = 3
Finds or
(Did not perform composition)
Unable to produce a solution.
Item 5b
4 3 2 1
Finds
Finds
(subtraction done incorrectly) or reversed functions g(2) – f(0) = 4.
Finds
(subtracts the inputs)
Any other solution or no solution
Item 6
4 3 2 1
Chooses C Chooses D or A Chooses B Chooses E
16
2PA =
2
4
=
PA
AP 16=2
41 PA =
1))2(( =fg 3))2(( =fg 4)2( =g2)2( −=f
4)0()2( −=− gf 0)0()2( =− gf 2)0()2( =− gf
86
Item 7
4 3 2 1
States that the position of car A is ahead of car B
States that car A and car B are traveling at the same speed, but unable to determine position.
States that car A and car B are at the same position or that the cars traveled the same distance.
No solution given.
Item 8
4 3 2 1
Correctly assigns the solution as (1, 4) using appropriate notation
Includes 1 and 4 in solution.
Solution is only one x in the interval (1, 4) or interval for
Solution is not an x-value in (1, 4) or no solution given
Item 9
4 3 2 1
Defines a linear formula in the form
where a < 1.
Defines a linear formula in the form
where a ≥ 1.
Defines a linear formula in the form
No formula provided
Item 10
4 3 2 1
Successfully calculates
Calculates
with
some errors or
Calculate s(4) or s(1)
No Solution
)()( xgxf >
avt = avt = atv =
14)1()4(
−− ss 14
)1()4(−− ss
14)1()4(
−+ ss
87
Item 11a
4 3 2 1
Finds Finds Finds g(3)=0 or f(3)=-2
Unable to find a solution
Item 11b
4 3 2 1
Finds Finds Finds Any other value given.
Item 12
4 3 2 1
Produces
Produces Produces
Any other solution or no solution given
Item 13
4 3 2 1
Choice B Choice C or E Choice D Choice A
Item 14
4 3 2 1
Finds Finds but has errors in arithmetic
Finds correctly or with some errors in arithmetic
Does not produce a solution or unable to follow logic of how solution is produced.
4))3(( =gf 5))3(( =fg
2)1(1 =−−g 1)1(1 =−−f 3)1( =−g
ttf100
1)(1 =−
yt 100=
tf
10011 =−
25))2(( =hg ))2((hg ))2((gh
88
Item 15
4 3 2 1
Produces Produce where C ≠ 25
Produce some other formula relating A and s.
Unable to relate A and s.
Item 16
4 3 2 1
Choice B Choice D or E Choice A Choice C
Item 17
4 3 2 1
Choice B Choice A or E Choice D Choice C
Item 18
4 3 2 1
Interprets S(m + 12) as a horizontal shift 12 units left of S(m) or as the salary after m + 12 months on the job.
Interprets S(m + 12) as a vertical shift up 12 units of S(m) or 12 dollars more than the salary of S(m).
Comes to any other conclusion that involves S(m) and 12.
Unable to determine
225 sA π= 2sCA π=
89
Item 19
4 3 2 1
Determines the domain as the set of all x ≥ -2 and x ≠ 1
Determines the domain as the set of all x ≥ -2 or x ≠ 1
Determines the domain as x = -2 and/or x = 1
Makes no mention of either x = -2 or x = 1 playing a role in determination of domain (ie. Domain is the set of real numbers)
Item 20
4 3 2 1
Choice I and III Choice I or III Choice II or choice II and III or choice I and II
Chooses all 3 choices
Item 21
4 3 2 1
Chooses only choice II.
Chooses choice II along with one other choice
Chooses all 3 choices or choice I and II or choice I and III
Does not choose choice II.
Item 22
4 3 2 1
Choice II Choice II and III Choice I, II, and III Choice I or III only
90
Score Breakdown
Total Score Range
90-96 60-89 38-59 24-37 (with no 4’s)
Student is at the Process Level of working with functions
Student is at the Pre-Process Level of working with functions
Student is at the Action Level of working with functions
Student is at the Pre-Action Level of working with functions
91
APPENDIX D
Questionnaire 2
92
Questionnaire 2
Measuring Student Concept Image (Definition) of Function
Please determine if each of the models below represents a function. Please also give your reasoning as to why it is or is not.
1. for . function not a function (circle one)
Reason__________________________________________________________________
2. function not a function (circle one)
x f(x)
-2 3
0 3
-1 3
3 3
Reason__________________________________________________________________
3. The amount of money earned at a job. function not a function (circle one)
Reason__________________________________________________________________
4. function not a function (circle one)
Reason__________________________________________________________________
xxf ±=)( 0≥x
93
5. function not a function (circle one)
Reason__________________________________________________________________
6. If possible, create an example of a function which assigns to every number different from 0 its square and to 0 it assigns 1. If not possible, explain why not. (Vinner & Dreyfus, 1989)
________________________________________________________________________
7. If possible, create an example of a function all of whose values are equal to each other. If not possible, explain why not. (Vinner & Dreyfus, 1989)
________________________________________________________________________
8. Define what a function is:________________________________________________
________________________________________________________________________
94
APPENDIX E
RUBRIC for Questionnaire 2
95
RUBRIC for Questionnaire 2
Item 1
3 2 1
Identifies model as a function since every input (y) produces a single output (x) OR identifies model as not a function since every input (x) produces 2 outputs (y)
Identifies model is not a function and justification shows incomplete understanding of the definition of function (i.e. Vertical line test)
No reason or superficial reason given.
Item 2
3 2 1
Identifies model as a function since every input (x) produces a single output f(x)
Identifies model as a function but justifies with continuity or vertical line test
Identifies model as not a function or no justification given if states model is a function.
Item 3
3 2 1
Identifies model as a function since every person earns a specific amount of money or identifies as not a function since many people can earn the same amount.
Identifies model as a function but does not use people and money as input and output but rather uses x and y, but makes some connection to context
Identifies model as not a function or identifies model as a function but no reason or superficial reason given
96
Item 4
3 2 1
Identifies model as a function since every input produces a single output
Identifies model as a function but mentions continuity or vertical line test
Identifies model as not a function. Or no justification given.
Item 5
3 2 1
Identifies model as not a function since input -1 produces two outputs
Identifies model as a function with the reason that every input has one output. (Assumed student did not see the two arrows from -1)
No reason or superficial reason given.
Item 6
3 2 1
Able to produce a piece-wise function that satisfies the given conditions.
Produces a function that satisfies one of the given conditions
Unable to produce a function
Item 7
3 2 1
Able to produce a function that satisfies the condition
Produces a function (i.e. y = x) and explanation shows understanding of the definition of function
Unable to produce a function and/or explanation does not show complete understanding about function definition
97
Item 8
3 2 1
Defines function as a relation in which every input is mapped to exactly one output.
Defines a function as a rule, uses x as input and y as output, definition shows a general understanding of function, but is missing some information
Unable to define or makes mention of the vertical line test or continuity.
Strength of Definition
Strength Strong Average Weak
Score range 22-24 15-21 below 15
98
APPENDIX F
IRB Approval Form
99
100
APPENDIX G
Informed Consent/Assent Form
101
Informed Consent/Assent Study Title: Student understanding of functions and success in calculus Principal Investigator: Mr. Daniel Drlik Co-Investigator: Dr. Laurie Cavey Dear Parent/Guardian and Calculus Student: My name is Daniel Drlik and I am a student in the Mathematics Education Master’s Program at Boise State University. I am asking for your permission to include your child in my research. This consent will give you the information you will need to understand why this research study is being done and why your child is being invited to participate. It will also describe what your child will need to do to participate as well as any known risks, inconveniences or discomforts that your child may have while participating. I encourage you to ask questions at any time. If you decide to allow your child to participate, you will be asked to sign this form and it will be a record of your agreement to participate. You will be given a copy of this to keep. Purpose and Background
As a calculus teacher at the high school level, I often wonder why particular students are successful while others are not. To answer this question, I have designed two questionnaires which will be administered at the end of the year prior to entering calculus and the other at the beginning of the year in which you are taking calculus. Your child is being invited to participate because he/she will be a student in either my AP Calculus AB or my Introduction to Calculus during the 2014-2015 school year. Procedures
Your child will be asked to complete two questionnaires at the beginning of the school year in which they are enrolled in a Calculus course. Each of these questionnaires are designed to measure understanding of functions. I am asking for your permission to analyze these questionnaires for my research study. The results of these questionnaires will not be used as part of your child’s grade for either class. Your child’s participation will not require him/her to do anything above and beyond what he/she would be doing in class anyway. If you choose not to allow your child to participate, he/she will still complete the tasks, but I will not use the results of his/her questionnaires in my research report. All data will be coded by Dr. Cavey, the co-investigator, to ensure confidentiality prior to Mr. Drlik’s analysis. Risks/Discomforts
There are minimal risks associated with this study, as your child is not being asked to do anything that is not already part of either of his/her math courses. If, at any time, you do not wish for your child’s data to be analyzed for this research, you may withdraw your child’s participation. Extent of Confidentiality
Reasonable efforts will be made to keep the personal information in your child’s research record private and confidential. Any identifiable information obtained in connection with
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this study will remain confidential and will be disclosed only with your permission or as required by law. The members of the research team and the Boise State University Office of Research Compliance (ORC) may access the data. The ORC monitors research studies to protect the rights and welfare of research participants. Your child’s name will not be used in any written reports or publications which result from this research, unless you have given explicit permission for us to do this. Data will be kept for three years (per federal regulations) after the study is complete and then destroyed. Benefits
There will be no direct benefit to you or your child from participating in this study. However, the information gained from this research may help education professionals better understand what students understand in secondary mathematics classes. Payment
There will be no payment to you or your child as a result of taking part in this study. Questions
If you have any questions or concerns about participation in this study, you should first talk with the principal investigator at (208) 350-4340 or [email protected]. If you have questions about your or your child’s rights as a research participant, you may contact the Boise State University Institutional Review Board (IRB), which is concerned with the protection of volunteers in research projects. You may reach the board office between 8:00 AM and 5:00 PM, Monday through Friday, by calling (208) 426-5401 or by writing: Institutional Review Board, Office of Research Compliance, Boise State University, 1910 University Dr., Boise, ID 83725-1138. Participation in Research is Voluntary
You do not have to give your child permission to be in this study if you do not want to. If you volunteer your child to be in this study, you may withdraw from it at any time without consequences of any kind or loss of benefits to which you are otherwise entitled. Documentation of Consent I and my child have read this form and decided that I will allow my child to participate in the project described above. Its general purposes, the particulars of involvement and possible risks have been explained to my satisfaction. I understand I can withdraw my child at any time. ________________________________ ________________________________ Printed Name of Child Signature of Child
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_______________________________________ ________________________________ Printed Name of Parent/Guardian Signature of Parent/Guardian Date _______________________________________ ________________________________ Signature of Person Obtaining Consent Date
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